# One Sided Limits and Limits at Infinity

**Example** (Finding Horizontal Asymptotes) Find the horizontal asymptote of the
graph of the function

Solution . Dividing both numerator and denominator by x and using the properties
of limits, we have

Therefore, the line
is a horizontal asymptote. It is also important to
realize that

Therefore, the line
is another horizontal asymptote.

**Example** (Finding Horizontal Asymptotes) Find the horizontal asymptote of the
graph of the function

Solution . Dividing both numerator and denominator by x and using the properties
of limits, we have

Therefore, the line y = 1 is a horizontal asymptote. In computing the limit x→ -∞
we must remember that for x < 0, we

have
, so when we divide the numerator by x , when x < 0 we have,

Therefore, the horizontal asymptotes are y = ± 1.

**Exercises**

(1) Sketch the graph of the function

and then use the graph to determine which the following statements about the
function y = f(x) are true and which are

false?

(a) lim_{→0}+ f (x) = 1

(b) lim_{x→2} f (x) does not exist

(c) lim_{x→2} f (x) = 2

(d) lim_{x→1}- f (x) = 2

(e) lim_{x→1}+ f (x) = 1

(f) lim_{x→1} f (x) does not exist

(g) lim_{x→0}+ f (x) = lim_{x→0}- f (x)

(h) lim_{x→c} f (x) exists at every c in the open interval (-1, 1).

(i) lim_{x→c} f (x) exists at every c in the open interval (1, 3).

(j) limx_{→0}- f (x) = 0

(k) lim_{x→3}+ f (x) does not exist

(2) Sketch the graph of the function

and then use the graph to determine the following?

(a) Find lim_{x→2}+ f (x), lim_{x→2}- f (x), and f (2).

(b) Does lim_{x→2} f (x) exist? If so, what is it? If not, why not?

(c) Find lim_{x→-1}- f (x) and lim_{x→-1}+ f (x).

(d) Does lim_{x→-1} f (x) exist? If so, what is it? If not, why not?

(3) Let Use the graph of g to determine the following,

(a) Does lim_{x→0}+ g(x) exist? If so, what is it? If not, why not?

(b) Does lim_{x→0}- g(x) exist? If so, what is it? If not, why not?

(c) Does lim_{x→0} g(x) exist? If so, what is it? If not, why not?

(4) Graph .Find lim_{x→1}- f (x) and lim_{x→1}+ f (x). Does lim_{x→1} f (x) exist? If so, what is
it? If not, why

not?

(5) Graph . Find lim_{x→1}- f (x) and lim_{x→1}+ f (x). Does lim_{x→1} f (x) exist? If so, what is
it? If not,

why not?

(6) Graph .

(a) What is the domain and range of f ?

(b) At what points c, if any does lim_{x→c} f (x) exist?

(c) At what points does only the left-hand limit exist?

(d) At what points does only the right-hand limit exist?

(7) Find the one-sided limit algebraically ,

(8) Find the one-sided limit algebraically ,

(9) Find the one-sided limit algebraically ,

(10) Find the two -sided limit, where k is a constant.

(11) Find the two -sided limit,

(12) Find the two -sided limit,

(13) Find the two-sided limit, .

(14) Find the two-sided limit,

(15) Find the limit of the function as x→+∞ and x→-∞.

(16) Find the limit of the function as x→+∞ and x→-∞.

(17) Find the limit of the function as θ →+∞ and θ →-∞.

(18) Find the limit of the function sin x as x→+∞ and x→-∞.

(19) Find the limit of the function as x→+∞ and x→-∞.

(20) Find the limit of the function as x→+∞ and x→-∞.

(21) Find the limit of the function as x→+∞ and x→-∞.

(22) Find the limit of the function as x→+∞ and x→-∞.

(23) Find the limit of the function as x→+∞ and x→-∞.

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