Find lim x→0 sin2x/sin3x | Evaluate lim x→0 sin3x/sin2x

The limit of sin2x/sin3x when x approaches 0 is equal to 2/3, that is, lim_x→0 sin2x/sin3x = 2/3. The limit of sin3x/sin2x when x approaches 0 is equal to 3/2, that is, lim_x→0 sin3x/sin2x = 3/2.

The following formula will be useful to compute the above limits:

lim_x→0 sinx/x = 1 …(∗)

Limit of sin2x/sin3x when x approaches 0

The limit of sin2x/sin3x when x→0 can be computed as follows:

lim_x→0 ${\displaystyle \frac{\sin 2x}{\sin 3x}}$

= lim_x→0 ${\displaystyle \frac{\frac{\sin 2x}{2x}}{\frac{\sin 3x}{3x}}}\times {\displaystyle \frac{2}{3}}$

= ${\displaystyle \frac{2}{3}}$ lim_x→0 ${\displaystyle \frac{\frac{\sin 2x}{2x}}{\frac{\sin 3x}{3x}}}$

By the product rule of limits, the above limit

= ${\displaystyle \frac{2}{3}}$ lim_x→0 $\frac{\sin 2x}{2x}$ lim_x→0 ${\displaystyle \frac{1}{\frac{\sin 3x}{3x}}}$

Let z=2x and t=3x. Then both z, t →0 when x→0.

= ${\displaystyle \frac{2}{3}}$ lim_z→0 $\frac{\sin z}{z}$ lim_t→0 ${\displaystyle \frac{1}{\frac{\sin t}{t}}}$

= ${\displaystyle \frac{2}{3}}$ × 1 × ${\displaystyle \frac{1}{1}}$ by the above limit rule (∗).

= 2/3

Therefore, the limit of sin2x/sin3x when x→0 is 2/3.

ALSO READ:

limx→0 (ex-1)/x = 1	limx→∞ sinx/x = 0
limx→0 x/cosx = 0	limx→0 x/sinx = 1

Limit of sin3x/sin2x when x approaches 0

We evaluate the limit x→0 sin3x/sin2x as follows:

lim_x→0 ${\displaystyle \frac{\sin 3x}{\sin 2x}}$

= lim_x→0 ${\displaystyle \frac{\frac{\sin 3x}{3x}}{\frac{\sin 2x}{2x}}}\times {\displaystyle \frac{3}{2}}$

= ${\displaystyle \frac{3}{2}}$ lim_x→0 ${\displaystyle \frac{\frac{\sin 3x}{3x}}{\frac{\sin 2x}{2x}}}$

By the quotient rule of limits, we have the above limit

= ${\displaystyle \frac{3}{2}}$ ${\displaystyle \frac{\underset{x\to 0}{lim}\frac{\sin 3x}{3x}}{\underset{x\to 0}{lim}\frac{\sin 2x}{2x}}}$

Taking z=2x and t=3x as before, we have

= ${\displaystyle \frac{3}{2}}$ ${\displaystyle \frac{\underset{t\to 0}{lim}\frac{\sin t}{t}}{\underset{z\to 0}{lim}\frac{\sin z}{z}}}$

= ${\displaystyle \frac{3}{2}}$ × ${\displaystyle \frac{1}{1}}$ by the above limit formula (∗).

= 3/2

So the limit of sin3x/sin2x when x→0 is 3/2.

Have You Read These Limits?

lim_x→0 tanx/x = 1

lim_x→∞ tanx/x = undefined

lim_x→0 sin(1/x) = undefined

Epsilon – delta definition of limit

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