Using L’Hôpital’s Rule to Find a Limit

Find \(\lim\limits_{x\rightarrow 0}\dfrac{\tan x}{6x}\).

Solution We follow the steps for finding a limit using L’Hôpital’s Rule.

Step 1 Since \(\lim\limits_{x\rightarrow 0}\tan x=0\) and \(\lim\limits_{x\rightarrow 0}( 6x) =0\), the quotient \(\dfrac{\tan x }{6x}\) is an indeterminate form at \(0\) of the type \(\dfrac{0}{0}\).

Step 2 \(\dfrac{d}{\textit{dx}}\tan x=\sec ^{2}x\) and \(\dfrac{d}{\textit{dx}}\left( 6x\right) =6\).

Step 3 \(\lim\limits_{x\rightarrow 0}\dfrac{\dfrac{d}{\textit{dx}}\tan x}{\dfrac{d}{dx}\left( 6x\right) }=\) \(\lim\limits_{x\rightarrow 0}\dfrac{\sec ^{2}x}{6}=\dfrac{1}{6}.\)

It follows from L’Hôpital’s Rule that \(\lim\limits_{x\rightarrow 0}\dfrac{\tan x}{6x}=\dfrac{1}{6}\).