Question: "Find the limit.limx→0+xsin⁡x"

Anahi Solomon

Answered question

2022-03-31

Find the limit.

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

Jadyn Gentry

Beginner2022-04-01Added 12 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 L'Hospitals's 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.

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