Question

Evaluate the limit:
lim x→0 sin4x/3x​

Answer 1

$\displaystyle \sf \lim_{x \to 0} \frac{sin 4x}{3x} = \frac{4}{3}$

Given :

$\displaystyle \sf \lim_{x \to 0} \frac{sin 4x}{3x}$

To find :

Evaluate the limit

Formula Used :

$\displaystyle \sf \lim_{x \to 0} \frac{sin x}{x} = 1$

Solution :

Step 1 of 2 :

Write down the given limit

Here the given limit is

$\displaystyle \sf \lim_{x \to 0} \frac{sin 4x}{3x}$

Step 2 of 2 :

Evaluate the limit

$\displaystyle \sf \lim_{x \to 0} \frac{sin 4x}{3x}$

$\displaystyle \sf = \lim_{x \to 0} \frac{sin 4x}{4x} \times \frac{4}{3}$

Let u = 4x

$\displaystyle \sf Then \: as \: x \to \: 0 \: we \: have \: u \to 0$

$\displaystyle \sf = \lim_{u \to 0} \frac{sin u}{u} \times \frac{4}{3}$

$\displaystyle \sf = 1 \times \frac{4}{3} \: \: \: \bigg[ \: \because \:\lim_{x \to 0} \frac{sin x}{x} = 1 \bigg]$

$\displaystyle \sf = \frac{4}{3}$

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