Limit Laws
Quick Summary
• The limit laws are simple formulas that help us evaluate limits precisely.
• There is a concise list of the Limit Laws at the bottom of the page.
The Limit Laws
For the following equations, a and k are constants and n is an integer. Also,
assume lim f (x) and lim g(x) both exist.
x→a x→a
(1) Constant Law: lim k = k
x→a
This law deals with constant functions (horizontal lines). Since y is always
equal to k, it doesn’t matter what x approaches.
Example 1 lim 8 = 8
x→6
f (x) f (x)
10
y=8
8
y=k
6 k
2
x x
2 4 6 8 10 a
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(2) Identity Law: x→a
lim x = a
This law deals with the function y = x. Since y and x are equal, whatever
value x approaches, y will have to approach the same value.
Example 2 lim x = 4
x→4
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x
x
=
=
f (x) f (x)
y
5
3
a
2
1
x x
1 2 3 4 5 a
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(3) Addition Law: x→a
lim f (x) + g(x) = lim f (x) + lim g(x)
x→a x→a
This simply means, when we take the limit of an addition, we can just take
the limit of each term individually, then add the results.
Example 3 lim (x + 5).
x→−7
Solution:
lim (x + 5) = lim x + lim 5 Addition Law
x→−7 x→−7 x→−7
= −7 + 5 Identity Law and Constant Law
= −2
(4) Subtraction Law: lim f (x) − g(x) = lim f (x) − lim g(x)
x→a x→a x→a
Essentially the same as the Addition Law, but for subtraction. Work with
each term separately, then subtract the results.
Example 4 lim1 (x − 9) =
x→ 2
Solution:
lim (x − 9) = lim1 x − lim1 9 Subtraction Law
x→ 12 x→ 2 x→ 2
1
= −9 Identity and Constant Laws
2
1 18
= −
2 2
17
=−
2
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(5) Constant Coefficient Law: x→a
lim k · f (x) = k lim f (x)
x→a
If your function has a coefficient, you can take the limit of the function first,
and then multiply by the coefficient.
Example 5 lim (8x)
x→3
lim (8x) = 8 lim x Constant Coefficient Law
x→3 x→3
= 8(3) Identity Law
= 24
With the first 5 Limit Laws, we can now find limits of any linear function
that has the form y = mx + b.
Example 6 lim (4x − 3)
x→−2
lim (4x − 3) lim (4x) − lim 3 Subtraction Law
x→−2 x→−2 x→−2
= 4 lim x − lim 3 Constant Coefficient Law
x→−2 x→−2
= 4(−2) − 3 Identity and Constant Laws
= −8 − 3
= −11
(6) Multiplication Law: lim f (x) · g(x) = lim f (x) lim g(x)
x→a x→a x→a
If the function involves the product of two (or more) factors, we can just
take the limit of each factor, then multiply the results together.
Example 7 lim x2
x→5
lim x2 = lim (x · x)
x→5 x→5
= lim x lim x Multiplication Law
x→5 x→5
= (5)(5) Identity Law
= 25
The first 6 Limit Laws allow us to find limits of any polynomial function,
though Limit Law 7 makes it a little more efficient.
n
(7) Power Law: lim (f (x))n = lim f (x) provided lim f (x) 6= 0 if n < 0.
x→a x→a x→a
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When taking limits with exponents, you can take the limit of the function
first, and then apply the exponent. But you have to be careful! If the exponent
is negative, then the limit of the function can’t be zero!
Example 8 lim (x + 1)3
x→4
3
lim (x + 1)3 = lim (x + 1) Power Law
x→4 x→4
3
= lim x + lim 1 Addition Law
x→4 x→4
= (4 + 1)3 Identity and Constant Laws
3
=5
= 125
Example 9 lim (4x3 + 5x)
x→−2
lim (4x3 + 5x) = lim (4x3 ) + lim (5x) Addition Law
x→−2 x→−2 x→−2
= 4 lim (x3 ) + 5 lim x Constant Coefficient Law
x→−2 x→−2
3
= 4 lim x + 5 lim x Power Law
x→−2 x→−2
= 4(−2)3 + 5(−2) Identity Law
= 4(−8) − 10
= −32 − 10
= −42
lim f (x)
lim fg(x)
(8) Division Law: x→a (x)
= x→a
lim g(x) provided lim g(x) 6= 0.
x→a x→a
The Division Law tells us we can simply find the limit of the numerator and
the denominator separately, as long as we don’t get zero in the denominator.
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2x
Example 10 lim
x→12 x−4
2x lim (2x)
lim = x→12 Division Law
x→12 x − 4 lim (x − 4)
x→12
2 lim x
x→12
= Constant Coefficient Law
lim (x − 4)
x→12
2 lim x
x→12
= Subtraction Law
lim x − lim 4
x→12 x→12
2(12)
= Identity and Constant Laws
12 − 4
24
=
8
=3
With the first 8 Limit Laws, we can now find limits of any rational function.
p √
n
(9) Root Law: lim n
f (x) = L provided L > 0 when n is even.
x→a
For root functions, we can find the limit of the inside function first, and then
apply the root. We have to be careful that we don’t end up taking a square-root
of a negative number though!
√
Example 11 lim x + 18
x→−2
√ q
lim x + 18 = lim (x + 18) Root Law
x→−2 x→−2
q
= lim x + lim 18 Addition Law
x→−2 x→−2
√
= −2 + 18 Identity and Constant Laws
√
= 16
=4
(10) Composition Law
Suppose lim g(x) = M , where M is a constant. Also, suppose f is continu-
x→a
ous at M . Then
lim f (g(x)) = f lim g(x) = f (M ).
x→a x→a
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Example 12 x→π
lim sin(x).
lim sin(x) = sin lim x Composition Law
x→π x→π
= sin(π) Identity Law
=0
Example 13 lim ecos(πx)
x→3
lim ecos(πx) = elimx→3 cos(πx) Composition Law
x→3
= ecos(limx→3 (πx)) Composition Law
cos(π limx→3 x)
=e Constant Coefficient Law
= ecos(π(3)) Identity Law
cos(3π)
=e
= e−1
1
=
e
(11) Inequality Law
Suppose f (x) ≥ g(x) for all x near x = a. Then lim f (x) ≥ lim g(x).
x→a x→a
List of Limit Laws
Constant Law lim k = k
x→a
Identity Law lim x = a
x→a
Addition Law lim f (x) + g(x) = lim f (x) + lim g(x)
x→a x→a x→a
Subtraction Law lim f (x) − g(x) = lim f (x) − lim g(x)
x→a x→a x→a
Constant Coefficient Law lim k · f (x) = k lim f (x)
x→a x→a
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Multiplication Law lim f (x) · g(x) = lim f (x) lim g(x)
x→a x→a x→a
n
n
Power Law lim (f (x)) = lim f (x) provided lim f (x) 6= 0 if n < 0
x→a x→a x→a
lim f (x)
f (x) x→a
Division Law lim = provided lim g(x) 6= 0
x→a g(x) lim g(x)
x→a x→a
p √
Root Law lim n
f (x) = n L provided L > 0 when n is even.
x→a
Composition Law Suppose lim g(x) = M , where M is a constant. Also, sup-
x→a
pose f is continuous at M . Then lim f (g(x)) = f lim g(x) = f (M ).
x→a x→a
Inequality Law Suppose f (x) ≥ g(x) for all x near x = a. Then lim f (x) ≥
x→a
lim g(x).
x→a
Author: HT Goodwill
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