In class problems
Thursday, July 5th
Part I
Limit of a function: Definition (,δ).
1. Use the given graph of f (x) = x2 to find a number δ such that (Problem
24- section 1.3)
if |x − 1| < δ then x2 − 1 < 1
2
2. Prove the statement using the , δ definition of limit and illustrate with a
diagram like Figure 15. (Problem 30- section 1.3)
1
lim ( x + 3) = 2
2
x→−2
3. Prove the statement using the , δ definition of limit. (Problem 36- section
1.3)
x2 + x − 12
lim
=7
x→3
x−3
√
4. Find limt→0
t2 +9−3
t2
1
5. Find the limit, if it exists. If the limit does not exist explain why. (Problem
34- section 1.4)
2x + 12
lim
x→−6 |x + 6|
Part II
Properties of limits:
1. Find the limit
lim
x→0
2. Evaluate
lim
θ→0
tan(3x)
tan(5x)
cos(θ) − 1
θ
Try to convert it to sin(θ).
3. The graphs of f and g are given. Use them to evaluate each limit, if it
exists. If the limit does not exist, explain why.
(a) limx→2 [f (x) + g(x)]
(b) limx→1 [f (x) + g(x)]
(c) limx→0 [f (x)g(x)]
(x)
(d) limx→−1 [ fg(x)
]
(e) limx→2 [x3 f (x)]
p
(f) limx→1 3 + f (x)
4. Find the limit. (Problem 46- section 1.4)
sin2 (3t)
t→0
t2
lim
5.
lim
θ→0
sin(θ)
θ + tan(θ)
2
6.
lim x cot(x)
x→0
7. If p is a polynomial, show that limx→a p(x) = p(a).
Part III
Continuity I:
1. Explain why the function is discontinuous at x = 1. Sketch the graph of
the function. (Problem 14- section 1.5)
(
1
if x 6= 1
f (x) = x−1
2
if x = 1
2. From the graph of g state the intervals on which g is continuous. (Problem
4- section 1.5)
3. Explain why the function is discontinuous at a = 1. Sketch the graph of
the function.(Problem 13- section 1.5)
f (x) = −
1
(x − 1)2
4. (Problem 16- section 1.5)
(
f (x) =
x2 −x
x2 −1
1
3
if x 6= 1
if x = 1
Part IV
Continuity II:
1. Explain, using theorems you’ve learned why the function is continuous at
every number in its domain. State the domain.
F (x) =
2.
G(x) =
x
x2 + 5x + 6
√
3
x(1 + x3 )
3. Locate the discontinuities of the function and illustrate by graphing.
y=
4.
1
1 + sin(x)
√
y = tan( x)
5. Use continuity to evaluate the limit.
√
5+ x
lim √
x→−4
5+x
6.
lim sin(x + sin(x))
x→π
7. Which of the following functions f has a removable discontinuity at a?
If the discontinuity is removable, find a function g that agrees with f for
x 6= a and is continuous at a.
(a) f (x) =
(b) f (x) =
(c) f (x) =
(d) f (x) =
x2 −2x−8
, a=
x+2
x−7
|x−7| , a = 7
x3 +64
x+4 ,
√
3− x
9−x ,
−2
a = −4
a=9
Part V
Continuity III:
1. Prove that the equation has at least one real root. (Problem 44- section
1.5)
4
√
x−5=
1
x+3
Optional: Find the root up to three decimal places, using a graphing
device.
2. If f (x) = x2 + 10 sin(x), show that there is a number c such that f (c) =
1000.
3. Prove that the equation has at least one real root.
cos(x) = x3
4.
x5 − x2 − 4 = 0
Part VI
Limits at infinity:
1. For the function g whose graph is given, state the following.
(a) limx→∞ g(x)
(b) limx→−∞ g(x)
(c) limx→3 g(x)
(d) limx→0 g(x)
(e) limx→−2+ g(x)
(f) The equations if the asymptotes
2. Find a formula for a function that has vertical asymptotes x = 1 and
x = 3 and horizontal asymptote y = 1. (problem 38, section 1.6)
5
3. Sketch the graph of an example of a function f that satisfies all of the
given conditions.
lim f (x) = −∞, lim f (x) = ∞, lim f (x) = 0, lim+ f (x) = ∞, lim− f (x) = −∞
x→2
x→∞
x→−∞
x→0
x→0
4. Determine limx→1− x31−1 and limx→1+ x31−1 , by evaluating f (x) =
for values of x that approach 1 from the right and the left.
5. Find the limit
x + x3 + x5
x→∞ 1 − x2 + x4
lim
6
1
x3 −1