Solved example of limits of exponential functions
Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$
Multiplying the fraction by $\ln\left(1+3\sin\left(x\right)\right)$
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 $\lim_{x\to 0}\left(\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
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
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 functions is the sum of the derivatives of each function
The derivative of the constant function ($1$) is equal to zero
The derivative of a function multiplied by a constant ($3$) 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 ${f'(x) = \cos(x)\cdot D_x(x)}$
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, 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 $1$
Simplifying, we get
Calculate the power $e^{3}$
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