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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 lefthand limit exist?
(d) At what points does only the righthand limit exist?
(7) Find the onesided limit algebraically ,
(8) Find the onesided limit algebraically ,
(9) Find the onesided 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 twosided limit, .
(14) Find the twosided 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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