2.8. USING DERIVATIVES TO EVALUATE LIMITS
155
as we’ d like by taking x sufficiently close (but not equal) to a. We thus expand this notation
and language to include the possibility that either L or a can be ∞. For instance, for
f (x) =
1
x , we now write
lim
x→0 +
1
x
= ∞,
by which we mean that we can make
1
x as large as we like by taking x sufficiently close
(but not equal) to 0. In a similar way, we naturally write
lim
x→∞
1
x
= 0,
since we can make
1
x as close to 0 as we’ d like by taking x sufficiently large (i.e., by letting
x increase without bound).
In general, we understand the notation lim
x→a
f (x) = ∞ to mean that we can make f (x)
as large as we’ d like by taking x sufficiently close (but not equal) to a, and the notation
lim
x→∞
f (x) = L to mean that we can make f (x) as close to L as we’ d like by taking x
sufficiently large. This notation applies to left- and right-hand limits, plus we can also use
limits involving −∞. For example, returning to Figure 2.22 and f (x) =
1
x , we can say that
lim
x→0 −
1
x
= −∞ and lim
x→−∞
1
x
= 0.
Finally, we write
lim
x→∞
f (x) = ∞
when we can make the value of f (x) as large as we’ d like by taking x sufficiently large. For
example,
lim
x→∞
x
2 = ∞.
Note particularly that limits involving infinity identify vertical and horizontal asymptotes
of a function. If lim x→a f (x) = ∞, then x = a is a vertical asymptote of f , while if
lim x→∞ f (x) = L, then y = L is a horizontal asymptote of f . Similar statements can be
made using −∞, as well as with left- and right-hand limits as x → a − or x → a + .
In precalculus classes, it is common to study the end behavior of certain families of
functions, by which we mean the behavior of a function as x → ∞ and as x → −∞. Here
we briefly examine a library of some familiar functions and note the values of several
limits involving ∞.
For the natural exponential function e x , we note that lim x→∞ e x = ∞ and lim x→−∞ e x =
0, while for the related exponential decay function e −x , observe that these limits are reversed, with lim x→∞ e −x = 0 and lim x→−∞ e −x = ∞. Turning to the natural logarithm
function, we have lim x→0 + ln(x) = −∞ and lim x→∞ ln(x) = ∞. While both e x and ln(x)
grow without bound as x → ∞, the exponential function does so much more quickly than
155
as we’ d like by taking x sufficiently close (but not equal) to a. We thus expand this notation
and language to include the possibility that either L or a can be ∞. For instance, for
f (x) =
1
x , we now write
lim
x→0 +
1
x
= ∞,
by which we mean that we can make
1
x as large as we like by taking x sufficiently close
(but not equal) to 0. In a similar way, we naturally write
lim
x→∞
1
x
= 0,
since we can make
1
x as close to 0 as we’ d like by taking x sufficiently large (i.e., by letting
x increase without bound).
In general, we understand the notation lim
x→a
f (x) = ∞ to mean that we can make f (x)
as large as we’ d like by taking x sufficiently close (but not equal) to a, and the notation
lim
x→∞
f (x) = L to mean that we can make f (x) as close to L as we’ d like by taking x
sufficiently large. This notation applies to left- and right-hand limits, plus we can also use
limits involving −∞. For example, returning to Figure 2.22 and f (x) =
1
x , we can say that
lim
x→0 −
1
x
= −∞ and lim
x→−∞
1
x
= 0.
Finally, we write
lim
x→∞
f (x) = ∞
when we can make the value of f (x) as large as we’ d like by taking x sufficiently large. For
example,
lim
x→∞
x
2 = ∞.
Note particularly that limits involving infinity identify vertical and horizontal asymptotes
of a function. If lim x→a f (x) = ∞, then x = a is a vertical asymptote of f , while if
lim x→∞ f (x) = L, then y = L is a horizontal asymptote of f . Similar statements can be
made using −∞, as well as with left- and right-hand limits as x → a − or x → a + .
In precalculus classes, it is common to study the end behavior of certain families of
functions, by which we mean the behavior of a function as x → ∞ and as x → −∞. Here
we briefly examine a library of some familiar functions and note the values of several
limits involving ∞.
For the natural exponential function e x , we note that lim x→∞ e x = ∞ and lim x→−∞ e x =
0, while for the related exponential decay function e −x , observe that these limits are reversed, with lim x→∞ e −x = 0 and lim x→−∞ e −x = ∞. Turning to the natural logarithm
function, we have lim x→0 + ln(x) = −∞ and lim x→∞ ln(x) = ∞. While both e x and ln(x)
grow without bound as x → ∞, the exponential function does so much more quickly than