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Limits of Exponential Functions Calculator

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1

Solved example of limits of exponential functions

$\lim_{x\to0}\left(1+3sinx\right)^{\frac{1}{x}}$
2

Insert an $1$ multiplying the exponent

$\lim_{x\to0}\left(\left(1+3\sin\left(x\right)\right)^{1\left(\frac{1}{x}\right)}\right)$
3

Rewrite the $1$ as a division of $3$$\sin(x)$

$\lim_{x\to0}\left(\left(1+3\sin\left(x\right)\right)^{\frac{3\sin\left(x\right)}{3\sin\left(x\right)}\frac{1}{x}}\right)$
4

Multiply the $3$$\sin(x)$ in the exponent

$\lim_{x\to0}\left(\left(1+3\sin\left(x\right)\right)^{\frac{1}{3\sin\left(x\right)}\frac{3\sin\left(x\right)}{x}}\right)$
5

Rewriting the exponent

$\lim_{x\to0}\left(\left(\left(1+3\sin\left(x\right)\right)^{\frac{1}{3\sin\left(x\right)}}\right)^{\frac{3\sin\left(x\right)}{x}}\right)$
6

Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$

$\lim_{x\to0}\left(\left(1+3\sin\left(x\right)\right)^{\frac{1}{3\sin\left(x\right)}}\right)^{\lim_{x\to0}\left(\frac{3\sin\left(x\right)}{x}\right)}$
7

Applying the Sandwich Theorem, which states that: Let $I$ be an interval that contains the point $c$, and let $f(x)$, $g(x)$, and $h(x)$ be functions defined on $I$. If for every $x$ not equal to $c$ in the interval $I$ we have $g(x)\leq f(x)\leq h(x)$ and also suppose that: $\displaystyle\lim_{x\to c}{g(x)}=\lim_{x\to c}{h(x)}=L$, then: $\displaystyle\lim_{x\to c}{f(x)}=L$

$\lim_{x\to0}\left(\left(1+3\sin\left(x\right)\right)^{\frac{1}{3\sin\left(x\right)}}\right)^3$
8

Apply the limit rule of: $\displaystyle\lim_{x\to0}\left(1+x\right)^\frac{1}{x}=e$, where the $x$ represents the function $3\sin\left(x\right)$

$e^{3}$

Final Answer

$e^{3}$$\,\,\left(\approx 20.085536923187664\right)$

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