Get the answer to "Evaluate the following limits limx→0sin⁡7xsin⁡3x"

Dolly Robinson

Answered question

2020-11-10

Evaluate the following limits

lim_{x \to 0} \frac{\sin 7 x}{\sin 3 x}

casincal

Skilled2020-11-11Added 82 answers

We have to evaluate the limit:

lim_{x \to 0} \frac{\sin 7 x}{\sin 3 x}

We know that

lim_{x \to 0} \frac{\sin x}{x} = 1

lim_{x \to 0} \frac{\sin a x}{x}

in this case we need to multiply and divide by a since we do make same in the denominator as in the angle of sine.

lim_{x \to 0} \frac{\sin a x}{x} \times = lim_{x \to 0} \frac{\sin a x}{a x} \times a

= 1 \times a

= a

lim_{x \to 0} \frac{\sin a x}{\sin b x}

then same case will be for numerator and denominator.
hence,

lim_{x \to 0} \frac{\sin a x}{\sin b x} = \frac{a}{b}

Finding given limit:
here, a=7
b=3
Hence,

lim_{x \to 0} \frac{\sin 7 x}{\sin 3 x} = \frac{7}{3}

Second method:
Multiplying and dividing by 7x for numerator and by 3x for denominator, we get

lim_{x \to 0} \frac{\sin 7 x}{\sin 3 x} \times \frac{7 x}{7 x} \times \frac{3 x}{3 x} = lim_{x \to 0} \frac{\frac{\sin 7 x}{7 x}}{\frac{3 x}{3 x}} \times \frac{7 x}{1} \times \frac{1}{3 x}

= \frac{1}{1} \times \frac{7}{1} \times \frac{1}{3}

= \frac{7}{3}

Hence, value of the given limit is

\frac{7}{3}

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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