Evaluate the limit   lim(x-->pi+)  (cosx)/(x-pi)

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lfryerda | High School Teacher | (Level 2) Educator

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To evaluate this limit, we try direct substitution before anything else, and see that letting `x->pi^+` gives `-1/0`

which is undefined, so we know that the limit as `x->pi` does not exist.  However, the right side will either go to positive or negative infinity.

Let `x=pi+epsilon` where `epsilon` is a small real positive number.

Then `cos(pi+epsilon)=cos pi cos epsilon-sin pi sin epsilon`

which becomes:

`-cos epsilon<0`

Also, the denominator of the limit is:

`x-pi`

`=pi+epsilon-pi`

`=epsilon>0`

This means that the limit is a negative number, and so we have:

The right-sided limit is negative infinity.

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