Expert answers to "Find the limit. limx→0+xsin⁡x"

Burhan Hopper

Answered question

2020-11-23

Find the limit.

lim_{x \to 0^{+}} \frac{\sqrt{x}}{\sqrt{\sin x}}

Clelioo

Skilled2020-11-24Added 88 answers

Given:

lim_{x \to 0^{+}} \frac{\sqrt{x}}{\sqrt{\sin x}}

Solve:

lim_{x \to 0^{+}} \frac{\sqrt{x}}{\sqrt{\sin x}} = lim_{x \to 0^{+}} \sqrt{\frac{\frac{1}{\sin x}}{x}}

= \sqrt{\frac{1}{lim_{x \to 0^{+}} \frac{\sin x}{x}}}

= \sqrt{\frac{1}{lim_{x \to 0^{+}} \frac{\sin x}{x}}}

taking

lim_{x \to 0^{+}} \frac{\sin x}{x}

If substituting in the limit directly, it is reduced to the indeterminate form, so apply LHospitalss rule

lim_{x \to 0^{+}} \frac{\frac{d \sin x}{d x}}{\frac{d x}{d x}} = lim_{x \to 0^{+}} \frac{\cos x}{1} = \frac{1}{1} = 1

Equation (1) becomes

lim_{x \to 0^{+}} \sqrt{\frac{x}{\sin x}} = \sqrt{\frac{1}{1}} = 1

The solution is 1.

Jeffrey Jordon

Expert2022-08-24Added 2605 answers

Answer is given below (on video)

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