The limit of sin x/x

The trick is to make the denominator match the inside of $\sin$. For $\frac{\sin 3x}{x}$, pull out a factor of 3 so the denominator becomes $3x$.

In general, for $a\neq 0$ and $b\neq 0$, $\lim_{x\to 0}\frac{\sin ax}{bx}=\frac{a}{b}$. The formula holds when angles are measured in radians.

Examples

1. 01What goes in □?$\displaystyle \frac{\sin 5x}{x} = 5 \times \frac{\sin 5x}{\square}$AnswerHide

   Answer$\displaystyle 5x$

   Steps

   1. Once the bottom matches the angle, the rule applies as is
   2. $\displaystyle 5 \times \frac{\sin 5x}{5x}$$\displaystyle = \frac{\sin 5x}{x}$
2. 02What is the value of this limit?$\displaystyle \lim_{x \to 0} \frac{\sin 3x}{x}$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. Make the bottom match the angle
   2. $\displaystyle \frac{\sin 3x}{x}$$\displaystyle = 3 \times \frac{\sin 3x}{3x}$
   3. The second factor approaches 1
   4. $\displaystyle 3 \times 1$$\displaystyle = 3$
3. 03What is the value of this limit?$\displaystyle \lim_{x \to 0} \frac{\sin 4x}{5x}$AnswerHide

   Answer$\displaystyle \frac{4}{5}$

   Steps

   1. Make the bottom match the angle
   2. $\displaystyle \frac{\sin 4x}{5x}$$\displaystyle = \frac{4}{5} \times \frac{\sin 4x}{4x}$
   3. The second factor approaches 1
   4. $\displaystyle \frac{4}{5} \times 1$$\displaystyle = \frac{4}{5}$
4. 04What is the value of this limit?$\displaystyle \lim_{x \to 0} \frac{\sin 2x}{6x}$AnswerHide

   Answer$\displaystyle \frac{1}{3}$

   Steps

   1. Make the bottom match the angle
   2. $\displaystyle \frac{\sin 2x}{6x}$$\displaystyle = \frac{1}{3} \times \frac{\sin 2x}{2x}$
   3. The second factor approaches 1
   4. $\displaystyle \frac{1}{3} \times 1$$\displaystyle = \frac{1}{3}$
5. 05What is the value of this limit?$\displaystyle \lim_{x \to 0} \frac{\sin 3x}{5x}$AnswerHide

   Answer$\displaystyle \frac{3}{5}$

   Steps

   1. Make the bottom match the angle
   2. $\displaystyle \frac{\sin 3x}{5x}$$\displaystyle = \frac{3}{5} \times \frac{\sin 3x}{3x}$
   3. The second factor approaches 1
   4. $\displaystyle \frac{3}{5} \times 1$$\displaystyle = \frac{3}{5}$
6. 06What goes in □?$\displaystyle \frac{\sin 2x}{x} = 2 \times \frac{\sin 2x}{\square}$AnswerHide

   Answer$\displaystyle 2x$

   Steps

   1. Once the bottom matches the angle, the rule applies as is
   2. $\displaystyle 2 \times \frac{\sin 2x}{2x}$$\displaystyle = \frac{\sin 2x}{x}$

FAQ

Q1Isn't $\lim_{x\to 0}\frac{\sin 3x}{x}$ also 1?AnswerHide

A

The limit is 1 only when the denominator matches the inside of $\sin$. Rewriting $\frac{\sin 3x}{x}=3\times\frac{\sin 3x}{3x}$ shows the limit is 3.

Q2Why must the angle be in radians?AnswerHide

A

In degrees, $\lim_{x\to 0}\frac{\sin x^\circ}{x}=\frac{\pi}{180}$, not 1. The rule $(\sin x)'=\cos x$ also holds only in radians.