Question

How do you find the limit of `(5x^2-8x-13)/(x^2-5)` as x approaches 3?

Answer 1

Use the properties of limits.

`lim_(xrarr3)(x^2-5)=lim_(xrarr3)x^2-lim_(xrarr3)5` (Subtraction property)
`=(lim_(xrarr3)x)^2-lim_(xrarr3)5` (Power property)
`=3^2-5` (Limits of Identity and constant functions)
`=9-5=4` (Arithmetic)

Because the limit of the denominator is not `0`, we can use the quotient property for limits.

By steps similar to those above, we can show that
`lim_(xrarr3)(5x^2-8x-13)=5(3)^2-8(3)-13=45-37=8`.

And, therefore `lim_(xrarr3)(5x^2-8x-13)/(x^2-5)=8/4=2`.