Limits at Infinity 3.5 On the agenda: 3 cases to find limits at infinity Other problems HW: p. 199-200 # 1-6, 9, 13-23 Odd

Let’s look at it graphically What is the limit as x approaches ?

Analytically What do you get when you plug in the c value? is another indeterminate form. Divide Every term by the highest power of x that you see. In this problem the highest exponential term is x2. Now plug in your c value.

Your Turn Do you see a relationship???? The limit as a function approaches infinity is the same as finding its Horizontal Asymptote!

Guidelines for Finding Limits at Infinity of Rational Functions If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0. If the degree of the numerator is equal to degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

Examples

Let’s look at:

Some Functions have 2 Horizontal Asymptotes

Analytically Divide everything by x which is equal to what under the radical? Now how do we get the other limit? Since we are going in the negative direction, for x < 0, you can write x =

Your Turn Find

Find: Example 2: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem:

Example 3: Find:

Often you can just “think through” limits. p