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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

Evaluate the limit $\lim_{x\to0}\left(\left(1+3\sin\left(x\right)\right)^{\frac{1}{x}}\right)$ by replacing all occurrences of $x$ by $0$

$\left(1+3\sin\left(0\right)\right)^{\frac{1}{0}}$

The sine of $0$ equals $0$

$\left(1+3\cdot 0\right)^{\frac{1}{0}}$

Multiply $3$ times $0$

$\left(1+0\right)^{\frac{1}{0}}$

Add the values $1$ and $0$

$1^{\frac{1}{0}}$

An expression divided by zero tends to infinity

$1^{\infty }$
3

Simplifying, we get

$1^{\infty }$
4

Apply the formula for limits that result in the indeterminate form $1^{\infty}$, which is as follows: $\lim_{x\to a}f(x)^{g(x)}=\lim_{x\to a}e^{\left[g(x)\cdot\left(f(x)-1\right)\right]}$

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

Multiplying the fraction by $3\sin\left(x\right)$

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

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

${\left(\lim_{x\to0}\left(e\right)\right)}^{\lim_{x\to0}\left(\frac{3\sin\left(x\right)}{x}\right)}$
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: $\displaystyle\lim_{x\to c}{f(x)}=L$

${\left(\lim_{x\to0}\left(e\right)\right)}^{3}$
8

The limit of a constant is just the constant

$e^{3}$

Final Answer

$e^{3}$

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