# Questions tagged [limits-without-lhopital]

The evaluation of limits without the usage of L'Hôpital's rule.

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### A Question on limits from differential calculus for beginners

$\lim_{x\rightarrow 0} \left(\frac{\sin x}{x}\right)^{\frac{1}{x^3}}$ Could anyone please tell me how to solve this using the Maclaurin series?
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### Prove that the limit does not exist: $\lim_{(x,y)\to(1,3)}\frac{|\sin(3x-y)|}{\sqrt{2(x-1)^2+5(y-3)^4}}$ (Without L'Hopital's)

It seems that the function is underfined at $(1,3)$, so when I try to directly plug in $(1,3)$ via limit laws, I get an undetermined term $\frac{0}{0}$. So I tried instead to consider individual paths ...
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### Limit of an exponential function can not use L'Hopital rule [closed]

I am stuck on a question about limits, and I must not use L'Hôpital rule. $$\lim_{x\to0}=\frac{9^x-5^x}{4^x-3^x}$$ I don't have any idea to solve this. Can anyone help me please? I appreciate it a lot!...
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### Taylor series and integral [closed]

If we know Taylor's series of $csch^{2}(x)$ as below (please look at https://www.wolframalpha.com) $$csch^2x= {\frac{1}{x^2}-\frac{1}{3}+\frac{x^2}{15}-\frac{2x^4}{189}+\frac{x^6}{675}-...}$$ Can ...
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### Applying the limit definition of a derivative on a radical function $x^{2/3}$

I'm trying to find the derivative of the following using the limit definition of a derivative: $$f(x)=x^{2/3}.$$ I know that the derivative of $f(x)$ is $\frac23x^{-1/3}$ by the power rule, but I can'...
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### What is the importance of small angle approximation [closed]

I have seen many approximations like $$\tan x \approx x$$ $$\sin x \approx x$$ $$\cos x \approx 1-\dfrac{x^2}{2}$$ $$\dfrac{\cos^2(x)}{\sin(x)\tan(x)} \approx \dfrac{x^2}{4} + x^{-2} - 1$$ These are ...
1 vote
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