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Exercise

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

Step-by-step Solution

1

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

$\lim_{x\to0}\left(e^{\frac{1}{x}\ln\left(1+3\sin\left(x\right)\right)}\right)$
2

Multiplying the fraction by $\ln\left(1+3\sin\left(x\right)\right)$

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

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

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

$\frac{0}{0}$
6

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

$\lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(\ln\left(1+3\sin\left(x\right)\right)\right)}{\frac{d}{dx}\left(x\right)}\right)$
7

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

$e^{\lim_{x\to0}\left(\frac{3\cos\left(x\right)}{1+3\sin\left(x\right)}\right)}$
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 $0$

$e^{3}$

Final answer to the exercise

$e^{3}$

Exact Numeric Answer

$20.0855369$

Try other ways to solve this exercise

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  • Solve using limit properties
  • Solve using direct substitution
  • Solve the limit using factorization
  • Solve the limit using rationalization
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
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Main Topic: Limits by L'Hôpital's rule

In mathematics, and more specifically in calculus, L'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms.

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