# Limits by direct substitution Calculator

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### Difficult Problems

1

Example

$\lim_{x\to∞}\left(e^x-\frac{1}{e^x}\right)^{\frac{1}{x^2}}$
2

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\left(\lim_{x\to∞}\left(-\frac{1}{e^x}\right)+\lim_{x\to∞}\left(e^x\right)\right)^{\left(\frac{1}{x^2}\right)}$
3

Evaluating the limit when $x$ tends to $∞$

$\left(\lim_{x\to∞}\left(e^x\right)-\frac{1}{e^∞}\right)^{\left(\frac{1}{x^2}\right)}$
4

Simplifying

$\left(\lim_{x\to∞}\left(e^x\right)-\frac{1}{e^∞}\right)^{\left(\frac{1}{x^2}\right)}$
5

Evaluating the limit when $x$ tends to $∞$

$\left(e^∞-\frac{1}{e^∞}\right)^{\left(\frac{1}{x^2}\right)}$
6

Simplifying

$\left(e^∞-\frac{1}{e^∞}\right)^{\left(\frac{1}{x^2}\right)}$
7

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

$\left(e^∞-\frac{1}{e^∞}\right)^{\left(x^{-2}\right)}$
8

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\left(e^∞-\frac{1}{e^∞}\right)^{\left(\frac{1}{x^{2}}\right)}$

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