Here, we show you a step-by-step solved example of limits of exponential functions. This solution was automatically generated by our smart calculator:
Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$
Multiplying the fraction by $\ln\left(1+3\sin\left(x\right)\right)$
Any expression multiplied by $1$ is equal to itself
Multiplying the fraction by $\ln\left(1+3\sin\left(x\right)\right)$
Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$
The limit of a constant is just the constant
Plug in the value $0$ into the limit
The sine of $0$ equals $0$
Multiply $3$ times $0$
Add the values $1$ and $0$
Calculating the natural logarithm of $1$
If we directly evaluate the limit $\lim_{x\to0}\left(\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'HĂ´pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
Find the derivative of the numerator
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 $\displaystyle f'(x)=\frac{a'}{a}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
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 ${f'(x) = \cos(x)\cdot D_x(x)}$
Multiplying the fraction by $3\cos\left(x\right)$
Find the derivative of the denominator
The derivative of the linear function is equal to $1$
Any expression divided by one ($1$) is equal to that same expression
After deriving both the numerator and denominator, and simplifying, the limit results in
Evaluate the limit $\lim_{x\to0}\left(\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}\right)$ by replacing all occurrences of $x$ by $0$
The sine of $0$ equals $0$
Multiply $3$ times $0$
Add the values $1$ and $0$
The cosine of $0$ equals $1$
Multiply $3$ times $1$
Divide $3$ by $1$
Evaluate the limit $\lim_{x\to0}\left(\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}\right)$ by replacing all occurrences of $x$ by $0$
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