Calculus- Early Transcendentals, 2021a

3.5. Infinite Limits and Limits at Infinity 87
Definition 3.16: Limit at Infinity (Formal Definition)
If f is a function, we say that lim
x→∞
f (x) =L if for every ε > 0 there is an N > 0 so that whenever
x > N, | f (x) − L| < ε. We may similarly define lim
x→−∞ f (x)=L.
We include this definition for completeness, but we will not explore it in detail. Suffice it to say that
such limits behave in much the same way that ordinary limits do; in particular there is a direct analog of
Theorem 3.8.
Example 3.17: Limit at Infinity
Compute lim
x→∞
2x 2 − 3x + 7
x 2 + 47x + 1 .
Solution. As x goes to infinity both the numerator and denominator go to infinity. We divide the numerator
and denominator by x 2 :
2x 2 − 3x + 7
lim
x→∞ x 2 + 47x + 1 = lim
x→∞
2 − 3 x + 7 x 2
1 + 47 x + 1 .
x 2
Now as x approaches infinity, all the quotients with some power of x in the denominator approach zero,
leaving 2 in the numerator and 1 in the denominator, so the limit again is 2.
♣
In the previous example, we divided by the highest power of x that occurs in the denominator in order
to evaluate the limit. We illustrate another technique similar to this.
Example 3.18: Limit at Infinity
Compute the following limit:
2x 2 + 3
lim
x→∞ 5x 2 + x .
Solution. As x becomes large, both the numerator and denominator become large, so it isn’t clear what
happens to their ratio. The highest power of x in the denominator is x 2 , therefore we will divide every term
in both the numerator and denominator by x 2 as follows:
2x 2 + 3
lim
x→∞ 5x 2 + x = lim 2 + 3/x 2
x→∞ 5 + 1/x .
Most of the limit rules from last lecture also apply to infinite limits, so we can write this as:
=
lim 2 + 3lim
x→∞ x→∞
lim 5 + lim
x→∞ x→∞
1
x 2
1
x
= 2 + 3(0)
5 + 0 = 2 5 .
1
1
Note that we used the theorem above to get that lim = 0and lim
x→∞ x x→∞ x 2 = 0.