201-103-RE - Calculus 1 WORKSHEET: LIMITS
201-103-RE - Calculus 1 WORKSHEET: LIMITS
1. Use the graph of the function f (x) to answer each question. Use , - or DN E where appropriate.
(a) f (0) = (b) f (2) = (c) f (3) = (d) lim f (x) =
x0-
(e) lim f (x) = x0
(f) lim f (x) = x3+
(g) lim f (x) = x3
(h) lim f (x) = x-
2. Use the graph of the function f (x) to answer each question. Use , - or DN E where appropriate.
(a) f (0) = (b) f (2) = (c) f (3) = (d) lim f (x) =
x-1
(e) lim f (x) = x0
(f) lim f (x) = x2+
(g) lim f (x) = x
3. Evaluate each limit using algebraic techniques. Use , - or DN E where appropriate.
x2 - 25
(a)
lim
x0
x2
- 4x - 5
x2 - 25
(b)
lim
x5
x2
- 4x - 5
7x2 - 4x - 3
(c)
lim
x1
3x2
- 4x + 1
x4 + 5x3 + 6x2
(d)
lim
x-2
x2(x + 1) - 4(x + 1)
3
(e) lim |x + 1| +
x-3
x
x+1-2 (f) lim
x3 x2 - 9
x2 + 7 - 3
(g) lim x3 x + 3
x2 + 2x - 8 (h) lim
x2 x2 + 5 - (x + 1)
2y2 + 2y + 4 1/3
(i) lim
y5
6y - 3
(j) lim 4 2 cos(x) - 5 x0
1
1
-
(k) lim 3 + x 3 - x
x0
x
2x + 8 1 (l) lim x2 - 12 - x
x-6 x + 6
(m) lim x2 - 2 - x2 + 1
x
(n) lim x - 2 - x
x-
(o) lim 6 2x - 14
x7
(p) lim 3 - 3x
x1-
x4 - 10
(q)
lim
x
4x3
+x
(r) lim 3 x - 3 x- 5 - x
3x3 + x2 - 2
(s)
lim
x
x2
+ x - 2x3 + 1
x+5 (t) lim
x 2x2 + 1
x5 + 1
(u) lim cos
x-
x6 + x5 + 100
2x (v) lim
x2 x2 - 4
3x
(w)
lim
x-1
x2
+ 2x + 1
x2 - 25
(x)
lim
x-1
x2
- 4x - 5
x2 - 5 + 2
(y) lim x3 x - 3
2x + sin(x)
(z) lim x0
x4
(A) lim 1 + ex2 x1- x - 1
(B) lim 2x2 - 3x
x
x+2- 2-x
(C) lim
x0
x
ex (D) lim
x0+ 1 + ln(x)
(E) lim x2 + 1 - 2x x 3x-1
(F) lim x1 x - 1
4. Find the following limits involving absolute values.
x2 - 1 (a) lim
x1 |x - 1|
(b) lim 1 + x2 x-2 |x + 2|
x2|x - 3| (c) lim
x3- x - 3
5. Find the value of the parameter k to make the following limit exist and be finite.
What is then the value of the limit?
x2 + kx - 20
lim
x5
x-5
6. Answer the following questions for the piecewise defined function f (x) described on the right hand side.
(a) f (1) = (b) lim f (x) =
x0
(c) lim f (x) = x1
sin(x)
f (x) =
2x2
for x < 1, for x > 1.
7. Answer the following questions for the piecewise defined function f (t) described on the right hand side.
(a) f (-3/2) = (b) f (2) = (c) f (3/2) = (d) lim f (t) =
t-2
(e) lim f (t) = t-1+
(f) lim f (t) = t2
(g) lim f (t) = t0
t2
t+6
f (t) =
t2 - t
3t
-
2
for t < -2 for - 1 < t < 2 for t 2
ANSWERS: 1. (a) DNE (b) 0 (c) 3 (d) - (e) DNE (f) 2 (g) DNE (h) 1 2. (a) 0 (b) DNE (c) 0 (d) DNE (e) 0 (f) - (g) 1 3.
(a) 5
(b)
5 3
(c) 5
(d) 1
(e) 1
(f )
1 24
(g)
1 6
(h) -18
(i)
4 3
(j) DNE
(k)
-
2 9
(l)
1 36
(m) 0
(n) DNE
(o) DNE
(p) 0
(q)
(r) -1
(s)
-
3 2
(t) 0
(u) 1
(v) DNE
(w) -
(x) DNE
(y) DNE
(z)
(A) -
(B) (C) 1
2
(D) 0
(E) -
(F)
2 3
4. (a) DNE (b) (c) -9
5. k = -1, limit is then equal to 9
6. (a) DNE (b) 0 (c) DNE
7. (a) DNE (b) 4 (c) 10 (d) DNE
8. (a) 0
(b) 0
(c)
5 3
(e)
5 2
(f) 4
(g) DNE
Pre-Calculus Rational functions worksheet
Name
For each of the rational functions find: a. domain b. holes c. vertical asymptotes d. horizontal asymptotes e. y-intercept f. x-intercepts
1.
f
x
x2 x2
x x
2 6
2.
f
x
2x2 x2 1
3. f x 3
x2
4. f x 2x 1
x
5.
f
x
x2 x 12 x2 9
6. f x x2 4
x3
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