Question

# Use l’Hospital’s rule to find $\lim _{x \rightarrow \infty}\left(1+\frac{c}{x}\right)^{x}$ where c is a constant.

Solution

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$\text{\underline {The L'Hospital's Rule}}$ is used when evaluating the limit that by substituting the limit value of a variable, goes to a form $\frac {0} {0}$ or $\frac {\infty } {\infty }$ (or some other, that can be reduced to these two). It says that if

$\lim\limits_{x \to a} \frac {f'(x) } {g'(x) } =L$

then

$\lim\limits_{x \to a} \frac {f(x)} {g(x) } =L$

To apply it, we differentiate the functions in the numerator and in the denominator and compute the limit $\lim\limits_{x \to a} \frac {f'(x) } {g'(x) }$. If it exists and equals to $L$, we conclude

$\lim\limits_{x \to a} \frac {f(x)} {g(x) } =L$

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