 # 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) limx→2 f (x) does not exist
(c) limx→2 f (x) = 2
(d) limx→1- f (x) = 2
(e) limx→1+ f (x) = 1
(f) limx→1 f (x) does not exist
(g) limx→0+ f (x) = limx→0- f (x)
(h) limx→c f (x) exists at every c in the open interval (-1, 1).
(i) limx→c f (x) exists at every c in the open interval (1, 3).
(j) limx→0- f (x) = 0
(k) limx→3+ f (x) does not exist

(2) Sketch the graph of the function and then use the graph to determine the following?
(a) Find limx→2+ f (x), limx→2- f (x), and f (2).
(b) Does limx→2 f (x) exist? If so, what is it? If not, why not?
(c) Find limx→-1- f (x) and limx→-1+ f (x).
(d) Does limx→-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 limx→0+ g(x) exist? If so, what is it? If not, why not?
(b) Does limx→0- g(x) exist? If so, what is it? If not, why not?
(c) Does limx→0 g(x) exist? If so, what is it? If not, why not?

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

(5) Graph . Find limx→1- f (x) and limx→1+ f (x). Does limx→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 limx→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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