Find lim _(xarrow 0)(cosx)^(1)/(x^(2))

Name: Find Lim _(xarrow 0)(cosx)^(1)/(x^(2))
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Question

Find lim _(xarrow 0)(cosx)^(1)/(x^(2))

Answer 1

To find the limit \(\lim_{x \to 0} (\cos x)^{\frac{1}{x^2}}\), we can use the property of limits involving exponentials and trigonometric functions. Let's rewrite the expression in a more manageable form.First, observe that:

Now, we need to find the limit of the exponent:

As

,

. We can use the Taylor series expansion for

around

:

Thus,

Therefore,

So, the limit becomes:

Thus, the final answer is:

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Find lim _(xarrow 0)(cosx)^(1)/(x^(2))

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