Calculus Ch.
1 FUNCTIONS AND LIMITS
1.4 Calculating Limits
이상준 교수
(경희대 수학과)
교재 : James Stewart,
Essential Calculus - Early Transcendentals (2 Ed)
1
Outline
(1) Computing limits with substitutions
(2) Trigonometric functions
(3) Computing limits with one-sided limits
(4) Squeeze theorem
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< Part 1 >
Computing limits
with substitutions
3
Limit Laws
We use the following properties of limits, called the Limit Laws.
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Limit laws
Sum Law
1. The limit of a sum is the sum of the limits.
Difference Law
2. The limit of a difference is the difference of the limits.
Constant Multiple Law
3. The limit of a constant times a function is
the constant times the limit of the function.
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Limit laws
Product Law
4. The limit of a product is the product of the limits.
Quotient Law
5. The limit of a quotient is the quotient of the limits
(provided that the limit of the denominator is not 0).
We can use the Product Law repeatedly.
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Limit Laws
Base cases:
If we now put 𝑓 (𝑥) = 𝑥 in Law 6 and use Law 8,
we get another useful special limit.
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Limit Laws
A similar limit holds for roots as follows.
More generally, we have the following law.
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Example 1
Evaluate the following limits and justify each step.
% ! '$% " &(
(a) lim(2x $ − 3x + 4) (b) lim
!→# %→&$ #&)%
Solution:
(a) lim (2x $ − 3x + 4) = lim (2x $ ) − lim 3x + lim 4 (by Laws 2 and 1)
!→# !→# !→# !→#
= 2 lim x $ − 3 lim x + lim 4 (by 3)
!→# !→# !→#
=2 5$ −3 5 +4 (by 9, 8 and 7)
= 39
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Example 1 – Solution
(b) We start by using Law 5, but its use is fully justified only at the final
stage when we see that the limits of the numerator and denominator exist
and the limit of the denominator is not 0.
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Direct Substitution Property
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Note
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< Part 2 >
Limits
of trigonometric functions
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Trigonometric functions
The trigonometric functions also enjoy the Direct Substitution Property.
Definition: the coordinates of the point 𝑃 in Figure are (cos q, sin q).
As q ® 0, 𝑃 approaches (1, 0).
So
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Direct Substitution Property
Recall that
Since cos 0 = 1 and sin 0 = 0, the equations in 1 assert that
the cosine and sine functions satisfy the Direct Substitution Property at 0.
The addition formulas for cosine and sine
can then be used to deduce that
these functions satisfy Direct Substitution.
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Calculating Limits
In other words,
This enables us to evaluate certain limits quite simply.
For example,
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Note
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< Part 3 >
𝟎/𝟎 cases
(when substitution is not good)
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Example 5
* " ',&)
Find lim * " .
*→+
Solution:
We can’t apply the Quotient Law immediately,
since the limit of the denominator is 0.
IDEA: Let’s find 𝑡 $ in the numerator and remove it!
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Example 5 – Solution
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Limits of trigonometric functions
On the basis of numerical and graphical evidence, we know that
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Example 11
./0 -&(
Evaluate lim .
-→+ -
Solution:
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Note
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< Part 4 >
Computing limits
using one-sided limits
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Theorem
Some limits are best calculated by finding the left- and right-hand limits.
The following theorem says that a two-sided limit exists
if and only if both of the one-sided limits exist and are equal.
When computing one-sided limits,
we use the fact that the Limit Laws
also hold for one-sided limits.
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Example 6
Show that lim 𝑥 = 0.
%→+
Solution:
We know that
Left-hand limit:
Right-hand limit:
Therefore,
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Example 8
The greatest integer function is defined by 𝑥 = the largest integer that
is less than or equal to 𝑥.
(
(For instance, −$ = −1.)
Show that lim 𝑥 does not exist.
%→)
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Example 8 – Solution
Graph:
Because these one-sided limits are not equal,
lim 𝑥 does not exist by Theorem 2.
%→)
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Note
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< Part 5 >
Squeeze theorem
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Squeeze theorem
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Example 9
(
Show that lim 𝑥 $ sin = 0.
%→+ %
Solution:
First note that we cannot use
(
because lim sin % does not exist.
%→+
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Example 9 – Solution
Instead, we apply the Squeeze Theorem.
We know that
Hence,
y = x2sin(1/x)
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Example 9 – Solution
We know that lim 𝑥 $ = 0 and lim(−𝑥 $ ) = 0
%→+ %→+
Taking 𝑓 𝑥 = −𝑥 $ , 𝑔 𝑥 = 𝑥 $ sin(1/𝑥) and ℎ 𝑥 = 𝑥 $
in Squeeze Theorem, we obtain
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Note
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