Resolved: "Evaluate the following limits. limx→0tan⁡7xsin⁡x"

Rivka Thorpe

Answered question

2021-02-19

Evaluate the following limits.

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

broliY

Skilled2021-02-20Added 97 answers

Given Function,

f (x) = \frac{\tan 7 x}{\sin x}

We have to find,

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

As we know,
If we get

\frac{0}{0}

an Indeterminate form then,

lim_{x \to 0} \frac{f (x)}{g (x)} = lim_{x \to 0} \frac{f^{'} (x)}{g^{'} (x)}

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

We have an indeterminate form of type

\frac{0}{0}

, we can apply L'Hospital Rule,

lim_{x \to 0} \frac{\tan 7 x}{\sin x} = lim_{x \to 0} \frac{\frac{d}{d x} (\tan 7 x)}{\frac{d}{d x} (\sin x)}

lim_{x \to 0} \frac{\tan 7 x}{\sin x} = lim_{x \to 0} \frac{7 \tan^{2} 7 x + 7}{\cos x}

as we know,

\tan^{2} x = \sec^{2} x - 1

lim_{x \to 0} \frac{\tan 7 x}{\sin x} = lim_{x \to 0} \frac{7 (\sec^{2} 7 x - 1) + 7}{\cos x}

= lim_{x \to 0} \frac{7 \sec^{2} 7 x - 7 + 7}{\cos x}

= lim_{x \to 0} \frac{7}{\cos x \times \cos^{2} 7 x}

put x=0

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

Result:

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

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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