Solved example of limits of exponential functions
Evaluate the limit $\lim_{x\to0}\left(\left(1+3\sin\left(x\right)\right)^{\frac{1}{x}}\right)$ by replacing all occurrences of $x$ by $0$
The sine of $0$ equals $0$
Multiply $3$ times $0$
Add the values $1$ and $0$
An expression divided by zero tends to infinity
Simplifying, we get
Apply the formula for limits that result in the indeterminate form $1^{\infty}$, which is as follows: $\lim_{x\to a}f(x)^{g(x)}=\lim_{x\to a}e^{\left[g(x)\cdot\left(f(x)-1\right)\right]}$
Multiplying the fraction by $3\sin\left(x\right)$
Apply the power rule of limits: $\displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}$
Applying the Sandwich Theorem, which states that: Let $I$ be an interval that contains the point $c$, and let $f(x)$, $g(x)$, and $h(x)$ be functions defined on $I$. If for every $x$ not equal to $c$ in the interval $I$ we have $g(x)\leq f(x)\leq h(x)$ and also suppose that: $\displaystyle\lim_{x\to c}{g(x)}=\lim_{x\to c}{h(x)}=L$, then: $\displaystyle\lim_{x\to c}{f(x)}=L$
The limit of a constant is just the constant
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