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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

Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$

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

Multiply the fraction and term

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

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

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

The limit of a constant is just the constant

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

Plug in the value $0$ into the limit

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

The sine of $0$ equals $0$

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

Multiply $3$ times $0$

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

Add the values $1$ and $0$

$\frac{\ln\left(1\right)}{0}$

Calculating the natural logarithm of $1$

$\frac{0}{0}$
6

If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
7

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

$\lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(\ln\left(1+3\sin\left(x\right)\right)\right)}{\frac{d}{dx}\left(x\right)}\right)$

Find the derivative of the numerator

$\frac{d}{dx}\left(\ln\left(1+3\sin\left(x\right)\right)\right)$

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{1+3\sin\left(x\right)}\frac{d}{dx}\left(1+3\sin\left(x\right)\right)$

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{1}{1+3\sin\left(x\right)}\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(3\sin\left(x\right)\right)\right)$

The derivative of the constant function ($1$) is equal to zero

$\frac{1}{1+3\sin\left(x\right)}\frac{d}{dx}\left(3\sin\left(x\right)\right)$

The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function

$3\left(\frac{1}{1+3\sin\left(x\right)}\right)\frac{d}{dx}\left(\sin\left(x\right)\right)$

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}$

Find the derivative of the denominator

$\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$1$
8

After deriving both the numerator and denominator, the limit results in

$e^{\lim_{x\to0}\left(\frac{\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}}{1}\right)}$
9

Any expression divided by one ($1$) is equal to that same expression

$e^{\lim_{x\to0}\left(\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}\right)}$
10

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$

$e^{3\lim_{x\to0}\left(\frac{\cos\left(x\right)}{1+3\sin\left(x\right)}\right)}$
11

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

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

The sine of $0$ equals $0$

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

Multiply $3$ times $0$

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

Add the values $1$ and $0$

$\frac{\cos\left(0\right)}{1}$

The cosine of $0$ equals $1$

$\frac{1}{1}$

Divide $1$ by $1$

$1$
12

Simplifying, we get

$1$
13

Multiply $3$ times $1$

$e^{3}$
14

Calculate the power $e^{3}$

$e^{3}$

Final Answer

$e^{3}$

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