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How should I calculate $\lim_{x\to 0} x\csc(x)+\frac{1}{x \csc(x)}$?

THOUGHTS:

I am not sure how to solve this problem. I tried breaking the problem down into two different limits:

$$ \lim_{x\to 0} x\csc(x)+\lim_{x\to 0}\frac{1}{x \csc(x)} $$

since if I can handle this two limits, then I can sum them together to get the desired result according to one of the limit properties.

However, when evaluating the limits as $x$ approaches $0$, I get an indeterminate form.

Can anyone help?

Andrew Chin
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Hannah
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1 Answers1

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We have that

$$x \csc x+\frac1{x \csc x} = \frac{x}{\sin x}+\frac1{\frac{x}{\sin x}}=\frac{x}{\sin x}+\frac{\sin x}{x}$$

then use that

user
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