Lim
x−→2 (x^2 − 4) tan(2x − 4) / (x − 2)^2

Find the following limits. Show all steps. If the limit does not exist, explain why
not? Not allow to use L’Hospitals rule.

Question

Lim
x−→2  (x^2 − 4) tan(2x − 4) / (x − 2)^2

Find the following limits. Show all steps. If the limit does not exist, explain why
not? Not allow to use L’Hospitals rule.

Aurora Alvarado · Accepted Answer

#lim_(x->2) ((x^2-4)tan(2x-4))/(x-2)^2 = 8#

Evaluate the limit:

#lim_(x->2) ((x^2-4)tan(2x-4))/(x-2)^2#

Note that:

#(x^2-4) = (x+2)(x-2)#

so we can simplify:

#lim_(x->2) ((x^2-4)tan(2x-4))/(x-2)^2 = lim_(x->2) ((x+2)(x-2)tan(2x-4))/(x-2)^2 = lim_(x->2) ((x+2)tan(2x-4))/(x-2)#

Multiply numerator and denominator by #2#:

#lim_(x->2) ((x^2-4)tan(2x-4))/(x-2)^2 = lim_(x->2) (2(x+2)tan(2x-4))/(2x-4)#

Now consider the limit:

#lim_(x->2) tan(2x-4)/(2x-4)#

Substituting #y=2x-4#, as #lim_(x->2) 2x-4 = 0# we have:

#lim_(x->2) tan(2x-4)/(2x-4) = lim_(y->0) tany/y#

that is a well known trigonometric limit:

#lim_(y->0) tany/y = lim_(y->0) 1/cosy siny/y = 1#

So:

#lim_(x->2) ((x^2-4)tan(2x-4))/(x-2)^2 = lim_(x->2) 2(x+2) * lim_(x->2) tan(2x-4)/(2x-4) = 8*1 = 8#

graph{((x^2-4)tan(2x-4))/(x-2)^2 [-4.873, 5.127, 6.26, 11.26]}