Evaluate the limit lim(x-->pi+) (cosx)/(x-pi)
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`
Also, the denominator of the limit is:
This means that the limit is a negative number, and so we have:
The right-sided limit is negative infinity.