Evaluating Limits Analytically Dividing out/ Cancellation Technique x2 − 1
lim
x→1 x − 1
Remember this limit from yesterday. We examined it yesterday and determined x3 − 1
lim
=2
that x→1 x − 1
Now lets see why. Completely Factor the numerator and denominator and cancel any common factors. Then attempt direct substitution AGAIN with the simplified function. Example x 2 − 3x + 2
lim
x→2
x − 2 Rationalization technique Rationalization technique can be recognized because of the presence of a radical plus or minus a number. x +1 −1
lim
x→0
x
As always try direct substitution first. If this does not work we will try to RATIONALIZE the function with the conjugate of the radical term Example 2+ x − 2
lim
x→0
x
Special Limits with Trig Functions sin x
1− cos x
lim
= 1 lim
= 0 x→0
x→0
x
x
Find the limit tan x
lim
x→0
x
sin 4x
lim
x→0
x