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# Find the limit of $\left(1+3\sin\left(x\right)\right)^{\frac{1}{x}}$ as $x$ approaches 0

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##  Final answer to the problem

$e^{3}$
Got another answer? Verify it here!

##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Solve using L'HÃ´pital's rule
• Solve without using l'HÃ´pital
• 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
Can't find a method? Tell us so we can add it.
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 $\lim_{x\to 0}\left(\frac{\ln\left(1+3\sin\left(x\right)\right)}{x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
6

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

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

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

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

$e^{\frac{3\cos\left(0\right)}{1+3\sin\left(0\right)}}$
9

The sine of $0$ equals $0$

$e^{\frac{3\cos\left(0\right)}{1+3\cdot 0}}$
10

Multiply $3$ times $0$

$e^{\frac{3\cos\left(0\right)}{1+0}}$
11

Add the values $1$ and $0$

$e^{\frac{3\cos\left(0\right)}{1}}$
12

The cosine of $0$ equals $1$

$e^{\frac{3\cdot 1}{1}}$
13

Multiply $3$ times $1$

$e^{\frac{3}{1}}$
14

Divide $3$ by $1$

$e^{3}$

##  Final answer to the problem

$e^{3}$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

###  Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.