2.4: The Limit Laws - Limits at Infinity (Lecture Notes)
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Finite Limits at Infinity and Horizontal Asymptotes
If the values of f(x) become arbitrarily close to the finite value L as x becomes sufficiently large, we say the function f has a finite limit at infinity and writelimx→∞f(x)=L.A similar definition holds for x→−∞.
If limx→∞f(x)=L or limx→−∞f(x)=L, we say the line y=L is a horizontal asymptote of f.
Revisiting the Limit Laws
Let f(x) and g(x) be defined for all x>a, where a is a real number. Assume that L and M are real numbers such that limx→∞f(x)=L and limx→∞g(x)=M. Let c be a constant. Then, each of the following statements holds:
- Sum and Difference Laws for Limits:limx→∞(f(x)±g(x))=limx→∞f(x)±limx→∞g(x)=L±M
- Constant Multiple Law for Limits:limx→∞cf(x)=c⋅limx→∞f(x)=cL
- Product Law for Limits:limx→∞(f(x)⋅g(x))=limx→∞f(x)⋅limx→∞g(x)=L⋅M
- Quotient Law for Limits:limx→∞f(x)g(x)=limx→∞f(x)limx→∞g(x)=LMfor M≠0.
- Power Law for Limits:limx→∞(f(x))n=(limx→∞f(x))n=Lnfor every positive integer n.
- Root Law for Limits:limx→∞n√f(x)=n√limx→∞f(x)=n√Lfor all L if n is odd and for L≥0 if n is even.
Each of these Limit Laws can be adjusted for x→−∞ as long as f(x) and g(x) are defined for all x<a, where a is a real number.
limx→∞x=∞,limx→−∞x=−∞,andlimx→±∞1x=0.
If p>0 is a rational number, thenlimx→∞1xp=0.If p>0 is a rational number such that xp is defined for all x, thenlimx→−∞1xp=0.
- Proof
- Let p be a rational number. The only constraint on using the Limit Laws is that the values of the limits must be finite. Therefore, a is allowed to be ±∞. Thus,
limx→∞1xp=limx→∞(1x)p=(limx→∞1x)p(a mixture of the Power Law and Root Law for Limits)=0p(by Lemma 2.4.1)=0
If we allow x→−∞, then additional restrictions on p must be made. Specifically, since p is a rational number, it cannot be equivalent to a simplified fraction with an even denominator. Otherwise, the even denominator would imply an even-indexed root of x, and this would result in imaginary numbers. Hence, as long as xp is defined,
limx→−∞1xp=0
by a similar derivation.
Evaluating Finite Limits at Infinity
Evaluate each of the following limits.
- limx→∞8.4x15−πx9+e4.2x15−x+1.228
- limx→−∞√11+2x+4x25x−8
- limx→∞(√1+x+16x2−4x)
Determine the horizontal asymptote(s) for the function.
- f(x)=e−x
- g(t)=tanh(t)
limx→∞tan−1(x)=π2,
limx→−∞tan−1(x)=−π2,
limx→∞tanh(x)=1,
limx→−∞tanh(x)=−1, and
limx→∞b−x=0, where b>1.
Revisiting the Squeeze Theorem
Let f(x),g(x), and h(x) be defined for all x>a, where a is a real number. If
f(x)≤g(x)≤h(x)
for all x>a and
limx→∞f(x)=L=limx→∞h(x)
where L is a real number, then limx→∞g(x)=L.
Likewise, let f(x),g(x), and h(x) be defined for all x<a, where a is a real number. If
f(x)≤g(x)≤h(x)
for all x<a and
limx→−∞f(x)=L=limx→−∞h(x)
where L is a real number, then limx→−∞g(x)=L.
Evaluate the limit.
- limx→∞e−3xsin(x)
- limx→∞e−3x+sin(x)
- limx→∞⌊x⌋x
Infinite Limits at Infinity
We say a function f has an infinite limit at infinity and write
limx→∞f(x)=∞.
if f(x) becomes arbitrarily large for x sufficiently large. We say a function has a negative infinite limit at infinity and write
limx→∞f(x)=−∞.
if f(x)<0 and |f(x)| becomes arbitrarily large for x sufficiently large. Similarly, we can define infinite limits as x→−∞.
- limx→∞x=∞
- limx→−∞x=−∞
- If p>0, limx→∞xp=∞
- If b>1, limx→∞bx=∞.
Evaluate the limit.
- limx→−∞2x+x2−x3+x8x5+8x4−x
- limx→∞ex
- limx→−∞(ax6+bx11), where a,b>0.