"Evaluating trig limit limx→01−cos⁡(x2)1−cos⁡(x) I have tried to..." was answered by our experts

Octavio Miller

Answered question

2022-01-29

Evaluating trig limit

lim_{x \to 0} \frac{\sqrt{1 - \cos (x^{2})}}{1 - \cos (x)}

I have tried to simplify the expression using the identity

1 - \cos (x) = 2 \sin^{2} (\frac{x}{2})

, but I have still failed to remove the indeterminate form.

Tyrn7i

Beginner2022-01-30Added 11 answers

Hint:

1 - \cos α = 2 \sin^{2} \frac{α}{2}

Apply this with both numerator and denominator. Then use

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

Dominique Green

Beginner2022-01-31Added 11 answers

A little more details:

lim_{x \to 0} \frac{\sqrt{1 - \cos (x^{2})}}{1 - \cos (x)}

= lim_{x \to 0} \frac{\sqrt{2 \sin^{2} (\frac{x^{2}}{2})}}{2 \sin^{2} \frac{x}{2}} =

= \frac{\sqrt{2}}{2} lim_{x \to 0} \frac{\sin (\frac{x^{2}}{2})}{\sin^{2} \frac{x}{2}}

= \sqrt{2} lim_{x \to 0} (\frac{\sin (\frac{x^{2}}{2})}{\frac{x^{2}}{2}} \frac{\frac{x^{2}}{4}}{\sin^{2} \frac{x}{2}})

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