Evaluate following limits.

(a)

(b)   

Solution

There in fact isn't a whole lot to this limit. In this case because there is only a 6 in the denominator we'll just factor out this and then employ the fact.

 (e)

This limit almost looks the similar as that actually in the sense that the argument of the sine is the similar as what is in the denominator.  Though, notice as well that, in the limit, value of x is going to 4 and not 0 as the fact requires.  Though, along with a change of variables we can illustrate that this limit is actually set to use the fact above regardless.

Hence, let θ = x -4 and then notice that as change of variable the limit becomes, x → 4 we have θ → 0 .  Thus, after doing the

All that is needed to use the fact is that the argument of the cosine is the same as the denominator.