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## Homework Statement

Evaluate: $$\lim_{θ \rightarrow 0} {\frac{1-cos θ}{sin θ}}$$

## Homework Equations

## The Attempt at a Solution

By using trigonometric identities, I get to:

$$\lim_{θ \rightarrow 0} {\frac{sin θ}{sin θ}}⋅\lim_{θ \rightarrow 0} {\frac{sin θ}{1+cos θ}}$$

By using the Limit Laws, I can evaluate the limit of the numerator and denominator of each factor individually, then multiply these limits together to get the limit of the given function:

$$\lim_{θ \rightarrow 0} {\frac{sin θ}{sin θ}} = \frac{0}{0}$$

$$\lim_{θ \rightarrow 0} {\frac{sin θ}{1+cos θ}}=\frac{0}{2}=0$$

Multiplying the two limits together, ##\frac{0}{0}⋅\frac{0}{2}=0##

Here is my confusion:

For ##\lim_{θ \rightarrow 0} {\frac{sin θ}{sin θ}}##, I am looking at this two ways. One is that I can use Limit Laws to evaluate the numerator and denominator separately, and their quotient is the limit of the whole thing. This gives ##\frac{0}{0}## which is undefined so that makes no sense.

But if I look at it as ##\frac{sin θ}{sinθ}=1##, this leaves me with 1⋅0=0 for the solution to the whole problem which does make sense.

So what's the deal with the Limit Law problem? Why do I get an undefined limit using them, whereas if I just take sin/sin=1, I get a defined answer?

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