Worksheet # 5: Limit Laws
1. Given lim
x→2f(x) = 5 and lim
x→2g(x) = 2, use limit laws (justify your work) to compute the follow-
ing limits. Note when working through a limit problem that your answers should be a chain of
equalities. Make sure to keep the lim
x→aoperator until the very last step.
(a) lim
x→22f(x)−g(x).
(b) lim
x→2
f(x)g(x)
x.
(c) lim
x→2f(x)2+x·g(x)2.
(d) lim
x→2[f(x)]3
2.
2. Calculate the following limits if they exist or explain why the limit does not exist.
(a) lim
x→1
x2−1
x−1
(b) lim
x→1
x2−1
x−2
(c) lim
x→2+
x2−1
x−2
(d) lim
x→9
x−9
√x−3
3. Find the value of csuch that lim
x→2
x2+ 3x+c
x−2exists. What is the limit?
4. Show that lim
h→0|h|
hdoes not exist by examining one-sided limits. Then sketch the graph of |h|
hand
check your reasoning.
5. True or false?
(a) The direct substitution property can always be used to compute limits.
(b) Let f(x) = (x+ 2)(x−1)
x−1and g(x) = x+ 2. Then f(x) = g(x).
(c) Let f(x) = (x+ 2)(x−1)
x−1and g(x) = x+ 2. Then lim
x→1f(x) = lim
x→1g(x).
(d) If both the one-sided limits of f(x) exist as xapproaches a, then lim
x→af(x) exists.
(e) Let p(x) = cnxn+cn−1xn−1+... +c1x+c0be a polynomial with coefficients cn, cn−1, ..., c0.
Then limx→ap(x) = cnan+cn−1an−1+... +c1a+c0.
(f) If lim
x→af(x) exists then limx→af(x) = f(a).