12.2 techniques for evaluating limits

HThere are three methods of evaluating limits:

1. Direct substitution 

  Ex : lim   X^3-10 = 17

         X->3

However, sometimes we get zero on denominator, which means undefined 

Then we use the following 2 methods

2. Cancellation

Ex lim (x^2+x-6)/(x+3)

    X->-3

= (x-2)(x+3)/(x+3)

=(x-2)

= -5

3. Rationalization

Ex lim ((x+1)^1/2-1)/x

    X->0

In this case, we rewrite the fraction by multiplying its conjugate pair

One -sided limit

Lim f(x) --> left 

X->c-

Lim f(x) --> right 

X->c+

Ex f(x)= { 4-x x>1

              4x-x^2 x<1 

Lim f(x) ?

X->1

Substitute x=1 we get the same answer from both sides , therefore the overall limit is 3