# is 0^0 indeterminate

This is not the same thing as saying 0 0 is undefined. The expression 00 \frac0000​ is not meaningful, so computing the limit requires another technique. x→0lim​xax​x→0lim​xxsin(x1​)​​=a  (a∈R)=DNE.​, Computing these limits, in general, is the fundamental problem of differential calculus since, as noted above, the derivative, f′(x)=lim⁡h→0f(x+h)−f(x)h This is where the subject of this section comes into play. - Indeterminate Forms," Convergence (July 2012), Mathematical Association of America □​​. Calculus textbooks also discuss the problem, usually in a section dealing with L'Hospital's Rule. The most common indeterminate forms stem from evaluating limits of a ratio of functions f(x)g(x)\frac{f(x)}{g(x)}g(x)f(x)​, namely 00\frac{0}{0}00​ and ∞∞\frac{\infty}{\infty}∞∞​. \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} \\ The strategy for evaluating exponential limits of the above types is to let y y y be the function which gives the indeterminate form and then to find the limit of ln⁡(y) \ln(y) ln(y). To see that the exponent forms are indeterminate note that He writes that it is understood that the base a of the logarithm should be a number greater than 1, thus avoiding his earlier reference to a possible problem with 00. \lim\limits_{x\to \infty} \left( 1+\dfrac4{x} \right)^x. \end{aligned} The first of these is a common misconception, since 00=1 0^0 = 1 00=1 in many contexts. Likewise if 0^0=1 then (0^0… lim⁡x→0+x1ln⁡x? \end{aligned} Let y=(1+4x)x y = \left( 1+\frac4{x} \right)^x y=(1+x4​)x. (So "0 times anything is 0" does not apply!) Log in here. New user? Confirm the limit has an indeterminate form. For instance, if lim⁡x→2f(x)=1 \lim\limits_{x\to 2} f(x) = 1 x→2lim​f(x)=1 and lim⁡x→2g(x)=3 \lim\limits_{x\to 2} g(x) = 3 x→2lim​g(x)=3, then lim⁡x→2(f(x)+g(x))=1+3=4\lim\limits_{x\to 2} \big(f(x)+g(x)\big) = 1+3 = 4 x→2lim​(f(x)+g(x))=1+3=4. Suppose we are given two functions, f(x) and g(x), with the properties that $$\lim_{x\rightarrow a} f(x)=0$$ and $$\lim_{x\rightarrow a} g(x)=0.$$ When attempting to evaluate [f(x)]g(x) in the limit as x approaches a, we are told rightly that this is an indeterminate form of type 00 and that the limit can have various values of f and g. This begs the question: are these the same? \begin{aligned} In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then it is said to assume an indeterminate form. Can we distinguish 00 as an indeterminate form and 00 as a number? What is. □ is the limit of an indeterminate form 00 \frac{0}{0} 00​. \lim_{x\to\infty} \ln(y) &= \lim_{x\to\infty} \ln\left( 1+\frac4{x} \right)^x \\ An indeterminate form 0 0 \frac{0}{0} 0 0 or ∞ ∞ \frac{\infty}{\infty} ∞ ∞ can have limit equal to any real number, or the limit may not exist. 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. \lim_{x\to 0} \dfrac{x\sin\left(\frac1x\right)}{x} &= \text{DNE}. If the denominator is positive, the limit is ∞ \infty ∞. &= \lim_{x\to 0^+} (-x) \\&= 0.\ _\square - Conclusion and Bibliography. Exponential: 0∞ 0^\infty 0∞ and ∞∞ \infty^\infty ∞∞ are not indeterminate; the limits are 0 0 0 and ∞ \infty ∞, respectively. Some of the arguments for why 0 0 0^0 0 0 is indeterminate or undefined are as follows: Argument 1: We know that a 0 = 1 a^0 = 1 a 0 = 1 (((for all a ≠ 0), a \ne 0), a = 0), but 0 a = 0 0^a = 0 0 a = 0 (((for all a > 0). As I said before, 0 0 can be considered indeterminate, but there are other possible values, and 0 0 = 1 is the most generally useful. The treatment of 0 0 has been discussed for several hundred years. a > 0). - Guglielmo Libri and Augustin Cauchy, What is 0^0? &= \lim_{x\to\infty} \dfrac{3+\dfrac7x}{\sqrt{1+\dfrac3x+\dfrac{7}{x^2}}+1} \\ 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? Understand the mathematics of continuous change. \end{aligned}x→∞lim​ln(y)​=x→∞lim​ln(1+x4​)x=x→∞lim​xln(1+x4​)=x→∞lim​x1​ln(1+x4​)​=x→∞lim​−x21​1+x4​1​(x2−4​)​=x→∞lim​1+x4​4​=4.​​(L’Hopital)​, So the original limit is e4 e^4 e4. f'(x) = \lim_{h\to 0} \dfrac{f(x+h)-f(x)}{h} f'(0).f′(0). &= \lim_{x\to 0^+} \dfrac{\frac1x}{\hspace{1mm} -\frac1{x^2}\hspace{1mm} } &&\text{(L'Hopital)} \\ If f(x) f(x) f(x) and g(x) g(x)g(x) are differentiable and f(x)g(x) \frac{f(x)}{g(x)} g(x)f(x)​ is one of these indeterminate forms, its limit can often be simplified using L'Hopital's rule.