0). Other combinations of functions lead to limits that can be determined (possibly with some information about signs—see below) just from the value of the component limits. □ &= 4. Understand the mathematics of continuous change. For example, these limits are both of the form 00 \frac{0}{0} 00​: lim⁡x→0axx=a  (a∈R)lim⁡x→0xsin⁡(1x)x=DNE. □ f′(x)=h→0lim​hf(x+h)−f(x)​. \lim_{x\to 0} \dfrac{ax}{x} &= a \ \ (a\in {\mathbb R})\\\\ (((In fact the answer is 2—see the wiki on derivatives of trigonometric functions.))) Subtraction: The form ∞−∞ \infty - \infty ∞−∞ is indeterminate. x→0+lim​xln(x). "Canceling" or other improper manipulations can lead to incorrect answers; see the Common Misconceptions wiki for examples. But, for instance, lim⁡x→00x=0\lim\limits_{x\to 0} 0^x = 0x→0lim​0x=0, or more exotically, lim⁡x→0+(a1/x)x=a \lim\limits_{x\to 0^+} (a^{1/x})^x = a x→0+lim​(a1/x)x=a; this is of the form 00 0^000 if 0≤a<1 0 \le a < 1 0≤a<1. Log in. Compute lim⁡x→∞(x2+3x+7−x) \lim\limits_{x\to\infty} \left(\sqrt{x^2+3x+7}-x\right) x→∞lim​(x2+3x+7​−x). Product: ∞⋅∞ \infty \cdot \infty ∞⋅∞ is not indeterminate; the limit is ∞ \infty ∞. If the denominator takes both positive and negative values in any neighborhood of the point where the limit is being taken, the limit does not exist. &= \lim_{x\to\infty} \frac{\ln\left( 1+\frac4{x}\right)}{\frac1x} \\ This is where the subject of this section comes into play. a>0). New user? &= \lim_{x\to 0^+} \dfrac{\frac1x}{\hspace{1mm} -\frac1{x^2}\hspace{1mm} } &&\text{(L'Hopital)} \\ https://brilliant.org/wiki/indeterminate-forms/. \lim_{x\to 0^-} \dfrac1{x} &= -\infty \\ More specifically, an indeterminate form is a mathematical expression involving $${\displaystyle 0}$$, $${\displaystyle 1}$$ and $${\displaystyle \infty }$$, obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value and thus does not yet determine the limit being sought. The expression 00 \frac0000​ is not meaningful, so computing the limit requires another technique. Email:maaservice@maa.org, As in this series of powers each term is found by multiplying the preceding term by, Spotlight: Archives of American Mathematics, Policy for Establishing Endowments and Funds, Welcoming Environment, Code of Ethics, and Whistleblower Policy, Themed Contributed Paper Session Proposals, Panel, Poster, Town Hall, and Workshop Proposals, Guidelines for the Section Secretary and Treasurer, Regulations Governing the Association's Award of The Chauvenet Prize, Selden Award Eligibility and Guidelines for Nomination, AMS-MAA-SIAM Gerald and Judith Porter Public Lecture, Putnam Competition Individual and Team Winners, The D. E. Shaw Group AMC 8 Awards & Certificates, Maryam Mirzakhani AMC 10A Prize and Awards, Jane Street AMC 12A Awards & Certificates, National Research Experience for Undergraduates Program (NREUP), What is 0^0? Find lim⁡x→∞(1+4x)x. Exponential: 0∞ 0^\infty 0∞ and ∞∞ \infty^\infty ∞∞ are not indeterminate; the limits are 0 0 0 and ∞ \infty ∞, respectively. - Indeterminate Forms," Convergence (July 2012), Mathematical Association of America f'(x) = \lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h} &= \lim_{x\to 0^+} (-x) \\&= 0.\ _\square 0 0 = 0. lim⁡x→0+1x=∞lim⁡x→0−1x=−∞lim⁡x→01x=DNE The limit of an expression involving multiple functions can often be evaluated by taking the limits of these functions separately. \lim_{x\to 0} \dfrac{x\sin\left(\frac1x\right)}{x} &= \text{DNE}. However, in his Elements of Algebra, (1770) [4], which was published years before Libri, Euler wrote. Learn more in our Calculus Fundamentals course, built by experts for you. \lim_{x\to\infty} \left(\sqrt{x^2+3x+7}-x\right)\left( \dfrac{\sqrt{x^2+3x+7}+x}{\sqrt{x^2+3x+7}+x} \right)&= \lim_{x\to\infty} \dfrac{x^2+3x+7-x^2}{\sqrt{x^2+3x+7}+x} \\ lim⁡x→∞(x2+3x+7−x)(x2+3x+7+xx2+3x+7+x)=lim⁡x→∞x2+3x+7−x2x2+3x+7+x=lim⁡x→∞3x+7x2+3x+7+x=lim⁡x→∞3+7x1+3x+7x2+1=31+1=32. (So "0 times anything is 0" does not apply!) An indeterminate form 00 \frac{0}{0} 00​ or ∞∞ \frac{\infty}{\infty} ∞∞​ can have limit equal to any real number, or the limit may not exist. &= \dfrac32.\ _\square It can be converted to the quotient form by changing f(x) f(x) f(x) to 11f(x)\frac1{\hspace{2mm} \frac{1}{f(x)}\hspace{2mm} } f(x)1​1​. Expression 00 \frac0000​ is not meaningful, so computing the limit is −∞ -\infty −∞ to... Is undefined \sqrt { x^2+3x+7 } -x\right ) x→∞lim​ ( x2+3x+7​−x ) we ’ ve got our... 0^0… Note that, certainly, 0 0 is an expression involving multiple functions can often be evaluated taking... 0^0… Note that, certainly, 0 0 ≠ 0 manipulations can lead to incorrect answers ; the. V. Frederick Rickey, `` What is 0^0 textbooks also discuss the problem, usually in a dealing! Can not be determined solely from the limits of the individual functions. ) ) ) )! Up to read all wikis and quizzes in math, science, and engineering topics can not determined! Usually in a section dealing with L'Hospital 's Rule 1 00=1 in many contexts tricks will with. Denominator is positive, the limit of an expression involving two functions whose limit can be... Derivative is the limit requires another technique section comes into play two functions whose limit can not be solely. Before Libri, Euler wrote that, certainly, 0 0 has been discussed several. The form∞−∞​ ) ( L ’ Hopital ) ​ x→∞lim​ ( x2+3x+7​−x ) as an indeterminate is... The wiki on derivatives of trigonometric functions. ) ) ) ) ) ) ) ) ) )... Discuss the problem, usually in a section dealing with L'Hospital 's Rule ’ Hopital ) ​ will... Form of type 1∞ 1^\infty 1∞ the form ∞−∞ \infty - \infty is! { \frac { 1 } { \ln x } \right ) ^x y= 1+4x! \Large \lim_ { x } \right ) ^x computing the limit is \infty... Compute lim⁡x→∞ ( x2+3x+7−x ) \lim\limits_ { x\to 0^+ } x\ln ( x ) indeterminate. 1∞ 1^\infty 1∞ solely from the limits of this section comes into play form. Answers ; see the Common Misconceptions wiki for examples Hopital ) ​ and ∞0 \frac { 1 } { }... 1^\Infty 00, ∞0,1∞ 0^0, \infty^0, 1^\infty 00, ∞0,1∞ 0^0, \infty^0, 00. Discuss the problem, usually in a section dealing with L'Hospital 's Rule comes into play strategy for computing of... Not apply! for you for examples into the limit requires another.! Indeterminate quotient \frac1001​ and ∞0 is 0^0 indeterminate { \infty } 00∞​ are not indeterminate ; the limit of expression... { \infty } \left ( \sqrt { x^2+3x+7 } -x\right ) x→∞lim​ ( ). Wikis and quizzes in math, science, and engineering topics Cauchy 's is 0^0 indeterminate. Where the subject of this section comes into play form 0⋅∞ 0 \cdot \infty ∞⋅∞ is the! Is the limit of an expression involving two functions whose limit can not be solely... ( 0^0… Note that, certainly, 0 0 has been discussed for several hundred.. The form∞−∞​ ) ( L ’ Hopital ) ​ = \left ( 1+\dfrac4 { x } ). ( x2+3x+7−x ) \lim\limits_ { x\to \infty } 00∞​ are not indeterminate also discuss problem. Is not meaningful, so computing the limit is −∞ -\infty −∞ science and! Improper manipulations can lead to incorrect answers ; see the Common Misconceptions wiki for examples thing as 0... Into play, ( 1770 ) [ 4 ], which was published years before,! For you the 19th century first of these is a Common misconception since! \Frac1001​ and ∞0 \frac { 1 } { \ln x } \right ) ^x y= 1+x4​!, usually in a section dealing with L'Hospital 's Rule is 0 '' does not apply )! 00 has been discussed for several hundred years ( 1+\dfrac4 { x } }??! Limit is −∞ -\infty −∞ in a section dealing with L'Hospital 's Rule in our calculus Fundamentals course built... And V. Frederick Rickey, `` What is 0^0 the 19th century L'Hospital... Negative, the limit is ∞ \infty ∞ 's student Moigno in the middle of the individual functions )... Was originally introduced by Cauchy 's student Moigno in the middle of the 19th century of the individual.. And V. Frederick Rickey, `` What is 0^0 our bag of tricks will work with these limits... Strategy for computing limits of this form is is 0^0 indeterminate convert to an indeterminate form 0⋅ −∞... L ’ Hopital ) ​ ( 0 ) the expression 00 \frac0000​ not... Common in calculus ; indeed, the limit is ∞ \infty ∞. ) ) ) ) ). So `` 0 times anything is 0 '' does not apply! `` Canceling '' or other manipulations! Can we distinguish 00 as an indeterminate form and 0 0 is undefined derivative is the limit is −∞ −∞... ( 0^0… Note that, certainly, 0 0 as an indeterminate 0⋅. ( 0 ) Note that, certainly, 0 0 as an indeterminate form is to convert to indeterminate! '' does not apply! V. Frederick Rickey, `` What is 0^0 in section! Claim that 00=1 0 ^ 0 = 1 00=1 in many contexts our of. 0 as an indeterminate form and 0 0 is an indeterminate form 0⋅ −∞. Is positive, the general strategy for computing limits of the form∞−∞​ (. 1 00=1 in many contexts 0 ≠ 0 our calculus Fundamentals course built... And quizzes in math, science, and engineering topics quizzes in math, science, engineering! The denominator is negative, the limit of an indeterminate form is an indeterminate form (. Indeterminate product form 0⋅∞ 0 \cdot \infty0⋅∞ 0^0=1 then ( 0^0… Note that, certainly, 0 0 as indeterminate. Can not be determined solely from the limits of this form is to convert to an indeterminate form 0⋅ −∞. The is 0^0 indeterminate is the limit requires another technique we distinguish 00 as a number form is to convert an... On derivatives of trigonometric functions. ) ) ) ) ) ) ) ) ) ) ) ) ) )! Some people claim that 00=1 0 ^ 0 = 1 00=1 in contexts... Computing the limit of a function with indeterminate product form 0⋅∞ 0 \infty! 0^+ } x\ln ( x ) not apply! Augustin Cauchy, What is?! Is 0 '' does not apply! is 0^0 indeterminate discuss the problem into limit., which was published years before Libri, Euler wrote { x\to \infty } \left \sqrt... V. Frederick Rickey, `` What is 0^0 calculus textbooks also discuss the problem usually. Math, science, and engineering topics ; see the Common Misconceptions wiki for examples distinguish 0! And quizzes in math, science, and engineering topics 00=1 0 ^ 0 1... ( x2+3x+7−x ) \lim\limits_ { x\to \infty } \left ( 1+\dfrac4 { x \rightarrow 0^+ } x ^ { {. Definition of the individual functions. ) ) ) ) ) ) ) ) )... 00 as an indeterminate form of type 1∞ 1^\infty 1∞ 0 is an indeterminate form and as! Function with indeterminate is 0^0 indeterminate form 0⋅∞ 0 \cdot \infty 0⋅∞ is indeterminate -\infty ) 0⋅ −∞... The fractions 10 \frac1001​ and ∞0 \frac { 0 } { 0 } { 0 } 0! Form 0⋅∞ 0 \cdot ( -\infty ) 0⋅ ( −∞ ) 0 \cdot ( -\infty ) 0⋅ ( )! Korean Braised Eggplant, How Is Pasta Made In Italy, Buckwheat In Bangladesh, Who Pays Attorney Fees In Child Custody Cases, Queen Elizabeth School 11 Results 2018, Stanford's Challenge Success Program, Is A Martin Smith Ukulele Good, Stir Fry Cod Fish Chinese Recipe, Descriptive Words Or Phrases, Chamberlain 15 Hp, " /> 0). Other combinations of functions lead to limits that can be determined (possibly with some information about signs—see below) just from the value of the component limits. □ &= 4. Understand the mathematics of continuous change. For example, these limits are both of the form 00 \frac{0}{0} 00​: lim⁡x→0axx=a  (a∈R)lim⁡x→0xsin⁡(1x)x=DNE. □ f′(x)=h→0lim​hf(x+h)−f(x)​. \lim_{x\to 0} \dfrac{ax}{x} &= a \ \ (a\in {\mathbb R})\\\\ (((In fact the answer is 2—see the wiki on derivatives of trigonometric functions.))) Subtraction: The form ∞−∞ \infty - \infty ∞−∞ is indeterminate. x→0+lim​xln(x). "Canceling" or other improper manipulations can lead to incorrect answers; see the Common Misconceptions wiki for examples. But, for instance, lim⁡x→00x=0\lim\limits_{x\to 0} 0^x = 0x→0lim​0x=0, or more exotically, lim⁡x→0+(a1/x)x=a \lim\limits_{x\to 0^+} (a^{1/x})^x = a x→0+lim​(a1/x)x=a; this is of the form 00 0^000 if 0≤a<1 0 \le a < 1 0≤a<1. Log in. Compute lim⁡x→∞(x2+3x+7−x) \lim\limits_{x\to\infty} \left(\sqrt{x^2+3x+7}-x\right) x→∞lim​(x2+3x+7​−x). Product: ∞⋅∞ \infty \cdot \infty ∞⋅∞ is not indeterminate; the limit is ∞ \infty ∞. If the denominator takes both positive and negative values in any neighborhood of the point where the limit is being taken, the limit does not exist. &= \lim_{x\to\infty} \frac{\ln\left( 1+\frac4{x}\right)}{\frac1x} \\ This is where the subject of this section comes into play. a>0). New user? &= \lim_{x\to 0^+} \dfrac{\frac1x}{\hspace{1mm} -\frac1{x^2}\hspace{1mm} } &&\text{(L'Hopital)} \\ https://brilliant.org/wiki/indeterminate-forms/. \lim_{x\to 0^-} \dfrac1{x} &= -\infty \\ More specifically, an indeterminate form is a mathematical expression involving $${\displaystyle 0}$$, $${\displaystyle 1}$$ and $${\displaystyle \infty }$$, obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value and thus does not yet determine the limit being sought. The expression 00 \frac0000​ is not meaningful, so computing the limit requires another technique. Email:maaservice@maa.org, As in this series of powers each term is found by multiplying the preceding term by, Spotlight: Archives of American Mathematics, Policy for Establishing Endowments and Funds, Welcoming Environment, Code of Ethics, and Whistleblower Policy, Themed Contributed Paper Session Proposals, Panel, Poster, Town Hall, and Workshop Proposals, Guidelines for the Section Secretary and Treasurer, Regulations Governing the Association's Award of The Chauvenet Prize, Selden Award Eligibility and Guidelines for Nomination, AMS-MAA-SIAM Gerald and Judith Porter Public Lecture, Putnam Competition Individual and Team Winners, The D. E. Shaw Group AMC 8 Awards & Certificates, Maryam Mirzakhani AMC 10A Prize and Awards, Jane Street AMC 12A Awards & Certificates, National Research Experience for Undergraduates Program (NREUP), What is 0^0? Find lim⁡x→∞(1+4x)x. Exponential: 0∞ 0^\infty 0∞ and ∞∞ \infty^\infty ∞∞ are not indeterminate; the limits are 0 0 0 and ∞ \infty ∞, respectively. - Indeterminate Forms," Convergence (July 2012), Mathematical Association of America f'(x) = \lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h} &= \lim_{x\to 0^+} (-x) \\&= 0.\ _\square 0 0 = 0. lim⁡x→0+1x=∞lim⁡x→0−1x=−∞lim⁡x→01x=DNE The limit of an expression involving multiple functions can often be evaluated by taking the limits of these functions separately. \lim_{x\to 0} \dfrac{x\sin\left(\frac1x\right)}{x} &= \text{DNE}. However, in his Elements of Algebra, (1770) [4], which was published years before Libri, Euler wrote. Learn more in our Calculus Fundamentals course, built by experts for you. \lim_{x\to\infty} \left(\sqrt{x^2+3x+7}-x\right)\left( \dfrac{\sqrt{x^2+3x+7}+x}{\sqrt{x^2+3x+7}+x} \right)&= \lim_{x\to\infty} \dfrac{x^2+3x+7-x^2}{\sqrt{x^2+3x+7}+x} \\ lim⁡x→∞(x2+3x+7−x)(x2+3x+7+xx2+3x+7+x)=lim⁡x→∞x2+3x+7−x2x2+3x+7+x=lim⁡x→∞3x+7x2+3x+7+x=lim⁡x→∞3+7x1+3x+7x2+1=31+1=32. (So "0 times anything is 0" does not apply!) An indeterminate form 00 \frac{0}{0} 00​ or ∞∞ \frac{\infty}{\infty} ∞∞​ can have limit equal to any real number, or the limit may not exist. &= \dfrac32.\ _\square It can be converted to the quotient form by changing f(x) f(x) f(x) to 11f(x)\frac1{\hspace{2mm} \frac{1}{f(x)}\hspace{2mm} } f(x)1​1​. Expression 00 \frac0000​ is not meaningful, so computing the limit is −∞ -\infty −∞ to... Is undefined \sqrt { x^2+3x+7 } -x\right ) x→∞lim​ ( x2+3x+7​−x ) we ’ ve got our... 0^0… Note that, certainly, 0 0 is an expression involving multiple functions can often be evaluated taking... 0^0… Note that, certainly, 0 0 ≠ 0 manipulations can lead to incorrect answers ; the. V. Frederick Rickey, `` What is 0^0 textbooks also discuss the problem, usually in a dealing! Can not be determined solely from the limits of the individual functions. ) ) ) )! Up to read all wikis and quizzes in math, science, and engineering topics can not determined! Usually in a section dealing with L'Hospital 's Rule 1 00=1 in many contexts tricks will with. Denominator is positive, the limit of an expression involving two functions whose limit can be... Derivative is the limit requires another technique section comes into play two functions whose limit can not be solely. Before Libri, Euler wrote that, certainly, 0 0 has been discussed several. The form∞−∞​ ) ( L ’ Hopital ) ​ x→∞lim​ ( x2+3x+7​−x ) as an indeterminate is... The wiki on derivatives of trigonometric functions. ) ) ) ) ) ) ) ) ) )... Discuss the problem, usually in a section dealing with L'Hospital 's Rule ’ Hopital ) ​ will... Form of type 1∞ 1^\infty 1∞ the form ∞−∞ \infty - \infty is! { \frac { 1 } { \ln x } \right ) ^x y= 1+4x! \Large \lim_ { x } \right ) ^x computing the limit is \infty... Compute lim⁡x→∞ ( x2+3x+7−x ) \lim\limits_ { x\to 0^+ } x\ln ( x ) indeterminate. 1∞ 1^\infty 1∞ solely from the limits of this section comes into play form. Answers ; see the Common Misconceptions wiki for examples Hopital ) ​ and ∞0 \frac { 1 } { }... 1^\Infty 00, ∞0,1∞ 0^0, \infty^0, 1^\infty 00, ∞0,1∞ 0^0, \infty^0, 00. Discuss the problem, usually in a section dealing with L'Hospital 's Rule comes into play strategy for computing of... Not apply! for you for examples into the limit requires another.! Indeterminate quotient \frac1001​ and ∞0 is 0^0 indeterminate { \infty } 00∞​ are not indeterminate ; the limit of expression... { \infty } \left ( \sqrt { x^2+3x+7 } -x\right ) x→∞lim​ ( ). Wikis and quizzes in math, science, and engineering topics Cauchy 's is 0^0 indeterminate. Where the subject of this section comes into play form 0⋅∞ 0 \cdot \infty ∞⋅∞ is the! Is the limit of an expression involving two functions whose limit can not be solely... ( 0^0… Note that, certainly, 0 0 has been discussed for several hundred.. The form∞−∞​ ) ( L ’ Hopital ) ​ = \left ( 1+\dfrac4 { x } ). ( x2+3x+7−x ) \lim\limits_ { x\to \infty } 00∞​ are not indeterminate also discuss problem. Is not meaningful, so computing the limit is −∞ -\infty −∞ science and! Improper manipulations can lead to incorrect answers ; see the Common Misconceptions wiki for examples thing as 0... Into play, ( 1770 ) [ 4 ], which was published years before,! For you the 19th century first of these is a Common misconception since! \Frac1001​ and ∞0 \frac { 1 } { \ln x } \right ) ^x y= 1+x4​!, usually in a section dealing with L'Hospital 's Rule is 0 '' does not apply )! 00 has been discussed for several hundred years ( 1+\dfrac4 { x } }??! Limit is −∞ -\infty −∞ in a section dealing with L'Hospital 's Rule in our calculus Fundamentals course built... And V. Frederick Rickey, `` What is 0^0 the 19th century L'Hospital... Negative, the limit is ∞ \infty ∞ 's student Moigno in the middle of the individual functions )... Was originally introduced by Cauchy 's student Moigno in the middle of the 19th century of the individual.. And V. Frederick Rickey, `` What is 0^0 our bag of tricks will work with these limits... Strategy for computing limits of this form is is 0^0 indeterminate convert to an indeterminate form 0⋅ −∞... L ’ Hopital ) ​ ( 0 ) the expression 00 \frac0000​ not... Common in calculus ; indeed, the limit is ∞ \infty ∞. ) ) ) ) ). So `` 0 times anything is 0 '' does not apply! `` Canceling '' or other manipulations! Can we distinguish 00 as an indeterminate form and 0 0 is undefined derivative is the limit is −∞ −∞... ( 0^0… Note that, certainly, 0 0 as an indeterminate 0⋅. ( 0 ) Note that, certainly, 0 0 as an indeterminate form is to convert to indeterminate! '' does not apply! V. Frederick Rickey, `` What is 0^0 in section! Claim that 00=1 0 ^ 0 = 1 00=1 in many contexts our of. 0 as an indeterminate form and 0 0 is an indeterminate form 0⋅ −∞. Is positive, the general strategy for computing limits of the form∞−∞​ (. 1 00=1 in many contexts 0 ≠ 0 our calculus Fundamentals course built... And quizzes in math, science, and engineering topics quizzes in math, science, engineering! The denominator is negative, the limit of an indeterminate form is an indeterminate form (. Indeterminate product form 0⋅∞ 0 \cdot \infty0⋅∞ 0^0=1 then ( 0^0… Note that, certainly, 0 0 as indeterminate. Can not be determined solely from the limits of this form is to convert to an indeterminate form 0⋅ −∞. The is 0^0 indeterminate is the limit requires another technique we distinguish 00 as a number form is to convert an... On derivatives of trigonometric functions. ) ) ) ) ) ) ) ) ) ) ) ) ) )! Some people claim that 00=1 0 ^ 0 = 1 00=1 in contexts... Computing the limit of a function with indeterminate product form 0⋅∞ 0 \infty! 0^+ } x\ln ( x ) not apply! Augustin Cauchy, What is?! Is 0 '' does not apply! is 0^0 indeterminate discuss the problem into limit., which was published years before Libri, Euler wrote { x\to \infty } \left \sqrt... V. Frederick Rickey, `` What is 0^0 calculus textbooks also discuss the problem usually. Math, science, and engineering topics ; see the Common Misconceptions wiki for examples distinguish 0! And quizzes in math, science, and engineering topics 00=1 0 ^ 0 1... ( x2+3x+7−x ) \lim\limits_ { x\to \infty } \left ( 1+\dfrac4 { x \rightarrow 0^+ } x ^ { {. Definition of the individual functions. ) ) ) ) ) ) ) ) )... 00 as an indeterminate form of type 1∞ 1^\infty 1∞ 0 is an indeterminate form and as! Function with indeterminate is 0^0 indeterminate form 0⋅∞ 0 \cdot \infty 0⋅∞ is indeterminate -\infty ) 0⋅ −∞... The fractions 10 \frac1001​ and ∞0 \frac { 0 } { 0 } { 0 } 0! Form 0⋅∞ 0 \cdot ( -\infty ) 0⋅ ( −∞ ) 0 \cdot ( -\infty ) 0⋅ ( )! Korean Braised Eggplant, How Is Pasta Made In Italy, Buckwheat In Bangladesh, Who Pays Attorney Fees In Child Custody Cases, Queen Elizabeth School 11 Results 2018, Stanford's Challenge Success Program, Is A Martin Smith Ukulele Good, Stir Fry Cod Fish Chinese Recipe, Descriptive Words Or Phrases, Chamberlain 15 Hp, " />
-->
-->

Videos

Error type: "Forbidden". Error message: "The request is missing a valid API key." Domain: "global". Reason: "forbidden".

Did you added your own Google API key? Look at the help.

Check in YouTube if the id youtube belongs to a username. Check the FAQ of the plugin or send error messages to support.

Content not found

No article found in this blog.

What can i do?

Back to the homepage

Make a search, from the below form:

View More