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We provide a proof of this theorem in the special case when $f,g,{f}^{\prime},$ and ${g}^{\prime}$ are all continuous over an open interval containing $a.$ In that case, since $\underset{x\to a}{\text{lim}}f\left(x\right)=0=\underset{x\to a}{\text{lim}}g\left(x\right)$ and $f$ and $g$ are continuous at $a,$ it follows that $f\left(a\right)=0=g\left(a\right).$ Therefore,
Note that L’Hôpital’s rule states we can calculate the limit of a quotient $\frac{f}{g}$ by considering the limit of the quotient of the derivatives $\frac{{f}^{\prime}}{{g}^{\prime}}.$ It is important to realize that we are not calculating the derivative of the quotient $\frac{f}{g}.$
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Evaluate each of the following limits by applying L’Hôpital’s rule.
Evaluate $\underset{x\to 0}{\text{lim}}\frac{x}{\text{tan}\phantom{\rule{0.1em}{0ex}}x}.$
$1$
We can also use L’Hôpital’s rule to evaluate limits of quotients $\frac{f\left(x\right)}{g\left(x\right)}$ in which $f\left(x\right)\to \text{\xb1}\infty $ and $g\left(x\right)\to \text{\xb1}\infty .$ Limits of this form are classified as indeterminate forms of type $\infty \text{/}\infty .$ Again, note that we are not actually dividing $\infty $ by $\infty .$ Since $\infty $ is not a real number, that is impossible; rather, $\infty \text{/}\infty .$ is used to represent a quotient of limits, each of which is $\infty $ or $\text{\u2212}\infty .$
Suppose $f$ and $g$ are differentiable functions over an open interval containing $a,$ except possibly at $a.$ Suppose $\underset{x\to a}{\text{lim}}f\left(x\right)=\infty $ (or $\text{\u2212}\infty )$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=\infty $ (or $\text{\u2212}\infty ).$ Then,
assuming the limit on the right exists or is $\infty $ or $\text{\u2212}\infty .$ This result also holds if the limit is infinite, if $a=\infty $ or $\text{\u2212}\infty ,$ or the limit is one-sided.
Evaluate each of the following limits by applying L’Hôpital’s rule.
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