TOPIC 1.
3: Illustration of Limit Theorems
In the following statements, c is a constant, and f and g are functions which may or may not
have c in their domains.
1. The limit of a constant is itself. If k is any constant, then
lim k = k.
x→c
For example,
(a) lim 2 = 2
x→c
(b) lim −3.14 = −3.14
x→c
(c) lim 789 = 789
x→c
2. The limit of x as x approaches c is equal to c. This may be thought of as the substitution
law because x is simply substituted by c.
lim x = c.
x→c
For example,
(a) lim x = 9
x→9
(b) lim x = 0.005
x→0.005
(c) lim x = −10
x→−10
For the remaining theorems, we will assume that the limits of f and g both exist as x
approaches c and that they are L and M , respectively. In other words,
lim f (x) = L, and lim g(x) = M.
x→c x→c
3. The Constant Multiple Theorem: This says that the limit of a multiple of a function is
simply that multiple of the limit of the function.
lim k · f (x) = k · lim f (x) = k · L.
x→c x→c
For example, if lim f (x) = 4, then
x→c
(a) lim 8 · f (x) = 8 · lim f (x) = 8 · 4 = 32.
x→c x→c
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(b) lim −11 · f (x) = −11 · lim f (x) = −11 · 4 = −44.
x→c x→c
3 3 3
(c) lim · f (x) = · lim f (x) = · 4 = 6.
x→c 2 2 x→c 2
4. The Addition Theorem: This says that the limit of a sum of functions is the sum of the
limits of the individual functions. Subtraction is also included in this law, that is, the limit
of a difference of functions is the difference of their limits.
lim ( f (x) + g(x) ) = lim f (x) + lim g(x) = L + M.
x→c x→c x→c
lim ( f (x) − g(x) ) = lim f (x) − lim g(x) = L − M.
x→c x→c x→c
For example, if lim f (x) = 4 and lim g(x) = −5 , then
x→c x→c
(a) lim (f (x) + g(x)) = lim f (x) + lim g(x) = 4 + (−5) = −1.
x→c x→c x→c
(b) lim (f (x) − g(x)) = lim f (x) − lim g(x) = 4 − (−5) = 9.
x→c x→c x→c
5. The Multiplication Theorem: This is similar to the Addition Theorem, with multiplication
replacing addition as the operation involved. Thus, the limit of a product of functions is
equal to the product of their limits.
lim (f (x) · g(x)) = lim f (x) · lim g(x) = L · M.
x→c x→c x→c
Again, let lim f (x) = 4 and lim g(x) = −5. Then
x→c x→c
lim f (x) · g(x) = lim f (x) · lim g(x) = 4 · (−5) = −20.
x→c x→c x→c
Remark 1: The Addition and Multiplication Theorems may be applied to sums, differences,
and products of more than two functions.
Remark 2: The Constant Multiple Theorem is a special case of the Multiplication Theorem.
Indeed, in the Multiplication Theorem, if the first function f (x) is replaced by a constant k,
the result is the Constant Multiple Theorem.
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6. The Division Theorem: This says that the limit of a quotient of functions is equal to the
quotient of the limits of the individual functions, provided the denominator limit is not equal
to 0.
f (x) lim f (x)
x→c
lim =
x→c g(x) lim g(x)
x→c
L
= , provided M 6= 0.
M
For example,
(a) If lim f (x) = 4 and lim g(x) = −5,
x→c x→c
f (x) lim f (x) 4 4
lim = x→c = =− .
x→c g(x) lim g(x) −5 5
x→c
(b) If lim f (x) = 0 and lim g(x) = −5,
x→c x→c
f (x) 0
lim = = 0.
x→c g(x) −5
f (x)
(c) If lim f (x) = 4 and lim g(x) = 0, it is not possible to evaluate lim , or we may
x→c x→c x→c g(x)
say that the limit DNE.
7. The Power Theorem: This theorem states that the limit of an integer power p of a function
is just that power of the limit of the function. If lim f (x) = L, then
x→c
lim (f (x))p = (lim f (x))p = Lp .
x→c x→c
For example,
(a) If lim f (x) = 4, then
x→c
lim (f (x))3 = (lim f (x))3 = 43 = 64.
x→c x→c
(b) If lim f (x) = 4, then
x→c
1 1
lim (f (x))−2 = (lim f (x))−2 = 4−2 = 2
= .
x→c x→c 4 16
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8. The Radical/Root Theorem: This theorem states that if n is a positive integer, the limit of
the nth root of a function is just the nth root of the limit of the function, provided the nth
root of the limit is a real number. Thus, it is important to keep in mind that if n is even,
the limit of the function must be positive. If lim f (x) = L, then
x→c
p
n
q √
n
lim f (x) = n lim f (x) = L.
x→c x→c
For example,
(a) If lim f (x) = 4, then
x→c
p q √
lim f (x) = lim f (x) = 4 = 2.
x→c x→c
p
(b) If lim f (x) = −4, then it is not possible to evaluate lim f (x) because then,
x→c x→c
q √
lim f (x) = −4,
x→c
and this is not a real number.
Solved Examples
EXAMPLE 1: Evaluate the following limits.
1. lim 0 (Ans. 0)
x→−1
2. lim −1 (Ans. -1)
x→32910
3. lim 1 (Ans. 1)
x→0.00001
EXAMPLE 2: Evaluate the given limits.
1. lim x (Ans. 1)
x→1
2. lim x (Ans. -0.01)
x→−0.01
3. lim x (Ans. 300)
x→300
EXAMPLE 3: Solve the following completely.
1. Given lim f (x) = −1, evaluate lim (5 · f (x)).
x→3 x→3
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2. Given lim g(x) = 2, determine lim (−2 · g(x)).
x→0 x→0
Solution. Using limit theorems, we have
1. lim (5 · f (x)) = 5 · lim f (x) = 5 · −1 = −5,
x→3 x→3
2. lim (−2 · g(x)) = −2 · lim g(x) = −2 · 2 = −4.
x→0 x→0
EXAMPLE 4: Use limit theorems to evaluate the following limits.
1. Determine lim (f (x) + g(x)) if lim f (x) = 2 and lim g(x) = −1.
x→1 x→1 x→1
2. Evaluate lim (f (x) − g(x)) given that lim f (x) = 0 and lim g(x) = −1.
x→−1 x→−1 x→−1
Solution. We use limit theorems to get
1. lim (f (x) + g(x)) = lim f (x) + lim g(x) = 2 + (−1) = 1,
x→1 x→1 x→1
2. lim (f (x) − g(x)) = lim f (x) − lim g(x) = 0 − (−1) = 1.
x→−1 x→−1 x→−1
EXAMPLE 5: Evaluate the following limits.
1. Given lim f (x) = 3 and lim g(x) = −1, determine lim f (x) · g(x).
x→2 x→2 x→2
f (x)
2. If lim f (x) = 9 and lim g(x) = −3, evaluate lim .
x→−3 x→−3 x→−3 g(x)
Solution. Using the theorems above, we have
1. lim f (x) · g(x) = lim f (x) · lim g(x) = 3 · (−1) = −3,
x→2 x→2 x→2
lim f (x)
f (x) x→−3 9
2. lim = = = −3.
x→−3 g(x) lim g(x) −3
x→−3
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EXAMPLE 6: Solve the following completely.
1. Evaluate lim (f (x))3 if lim f (x) = −1.
x→1 x→1
2. Evaluate lim (g(x))−2 if lim g(x) = 2.
x→0 x→0
Solution. Using the limit theorems, we obtain
1. lim (f (x))3 = ( lim f (x))3 = (−1)3 = −1,
x→1 x→1
1
2. lim (g(x))−2 = ( lim g(x))−2 = 2−2 = .
x→0 x→0 4
EXAMPLE 7: Evaluate the following limits.
p
1. Determine lim f (x) given that lim f (x) = 1.
x→−1 x→−1
p
2. Evaluate lim g(x) if lim g(x) = 9.
x→−2 x→−2
Solution. We use limit theorems to get
p q √
1. lim f (x) = lim f (x) = 1 = 1,
x→−1 x→−1
p q √
2. lim g(x) = lim g(x) = 9 = 3.
x→−2 x→−2
Supplementary Problems
1. Given lim f (x) = 3 and lim g(x) = −1, evaluate the following limits.
x→1 x→1
(a) lim 2 · f (x) (d) lim (f (x) − g(x))
x→1 x→1
(b) lim −3 · g(x) (e) lim (f (x) · g(x))
x→1
x→1
f (x)
(c) lim (f (x) + g(x)) (f) lim
x→1 x→1 g(x)
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2. Given lim f (x) = 2 and lim g(x) = −2, evaluate the following limits.
x→−1 x→−1
1 f (x)
(a) lim · f (x) (e) lim
x→−1 2 x→−1 g(x)
(b) lim (f (x) + g(x))
x→−1
(f) lim (f (x))2
(c) lim (f (x) − g(x)) x→−1
x→−1
(d) lim (f (x) · g(x)) (g) lim (g(x))−3
x→−1 x→−1
3. Given lim f (x) = 0 and lim g(x) = 1, evaluate the following limits.
x→0 x→0
(a) lim 139401 · f (x) f (x)
x→0 (e) lim
x→0 g(x)
(b) lim (f (x) + g(x))
x→0 p
(f) lim f (x)
(c) lim (f (x) − g(x)) x→0
x→0 p
(d) lim (f (x) · g(x)) (g) lim g(x)
x→0 x→0
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