Final answer to the problem
$e^{3}$
Got another answer? Verify it here!
Step-by-step Solution
Specify the solving method
Choose an option Solve using direct substitution Solve using L'H么pital's rule Solve using factorization Solve using rationalization Solve without using l'H么pital Suggest another method or feature
Send
1
Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$
$\lim_{x\to0}\left(e^{\frac{1}{x}\ln\left(1+3\sin\left(x\right)\right)}\right)$
Intermediate steps
2
Multiplying the fraction by $\ln\left(1+3\sin\left(x\right)\right)$
$\lim_{x\to0}\left(e^{\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}}\right)$
Explain this step further
3
Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$
${\left(\lim_{x\to0}\left(e\right)\right)}^{\lim_{x\to0}\left(\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}\right)}$
4
The limit of a constant is just the constant
$e^{\lim_{x\to0}\left(\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}\right)}$
Intermediate steps
5
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
$\frac{0}{0}$
Explain this step further
6
We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
$\lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(\ln\left(1+3\sin\left(x\right)\right)\right)}{\frac{d}{dx}\left(x\right)}\right)$
Intermediate steps
7
After deriving both the numerator and denominator, the limit results in
$e^{\lim_{x\to0}\left(\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}\right)}$
Explain this step further
8
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 $0$
$e^{\frac{3\cos\left(0\right)}{1+3\sin\left(0\right)}}$
$e^{\frac{3\cos\left(0\right)}{1+0}}$
10
Add the values $1$ and $0$
$e^{\frac{3\cos\left(0\right)}{1}}$
11
The cosine of $0$ equals
$e^{\frac{3}{1}}$
13
Calculate the power $e^{3}$
$e^{3}$
Final answer to the problem
$e^{3}$
Exact Numeric Answer
$20.085537$