13.4 Some Standard Limits:
- 1.
- \(\lim \limits _{x\longrightarrow 0} \frac {\sin x}{x}=1\)
- 2.
- \(\lim \limits _{x\longrightarrow 0} \frac {1-\cos x}{x}=0\)
- 1.
- \(\lim \limits _{x\longrightarrow 0} \frac {\sin 3x}{x}\)
- 2.
- \(\lim \limits _{x\longrightarrow 0} \frac {\tan x}{x}\)
- 3.
- \(\lim \limits _{x\longrightarrow 0} \frac {\sin x}{3x}\)
Solution.
\begin {align*} \lim _{x\longrightarrow 0} \frac {\sin 3x}{x} & = \lim _{x\longrightarrow 0} \frac {3\sin 3x}{3x}\\ & = 3 \lim _{x\longrightarrow 0} \frac {\sin 3x}{3x}\\ & = 3(1)\\ & = 3\\ \end {align*}
\begin {align*} \lim _{x\longrightarrow 0}\frac {\tan x}{x} & = \lim _{x\longrightarrow 0} \frac {\sin x}{x\cos x}\\ & = \lim _{x\longrightarrow 0} \left (\frac {1}{\cos x}\cdot \frac {\sin x}{x}\right )\\ & = \lim _{x\longrightarrow 0}\frac {1}{\cos x}\cdot \lim _{x\longrightarrow 0}\frac {\sin x}{x}\\ & = \frac {1}{\cos (0)}\cdot (1)\\ & = \frac {1}{1}\cdot (1)\\ & = 1\\ \end {align*}
Note 13.15. Finally, \(\lim \limits _{x\to 0}\frac {\sin x}{3x} =\frac {1}{3}\lim \limits _{x\to 0}\frac {\sin x}{x}=\frac {1}{3}\).
The pattern in all three is the same: \(\frac {\sin \square }{\square }\to 1\) only when the same expression stands above and below. In \(\frac {\sin 3x}{x}\) it does not, so the \(3x\) is manufactured by writing \(\frac {\sin 3x}{x}=3\cdot \frac {\sin 3x}{3x}\); in \(\frac {\sin x}{3x}\) the constant is simply taken outside. Reading \(\frac {\sin 3x}{x}\) as \(1\) is the standard mistake, and it comes from matching the pattern too loosely.
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