Given the function
, find the following limits without using your graphing calculator. Homework Help ✎
Before finding the limit, factor the numerator and the denominator. If the '0' in the denominator cancels out, then the limit exists.
For all limits as x →∞, you are looking for end behavior. For example, is there a horizontal asymptote? Or does the function approach +∞ or −∞ in the end? To answer this question, compare the highest power in the numerator with the highest power in the denominator.
Highest power in the numerator: x2
Highest power in the denominator: x2
Note: this means that f(x) has a horizontal asymptote of y = 1.
First, factor the expression. Visualize the left-hand and right-hand limits of the simplified function. Do they match up?
You can prove this by showing that
Limit does not exist.