Limits at a Point

7/30/2019 Limits at a Point

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74 CHAPTER 3. LIMITS

3.3 Limits at a point

Textbook pages 116-128

3.3.1 Definitions and examples

Rough definition:

Limits from the left and limits from the right: The particular value c that wewant to consider can be approached from the left or from the right, so we have to define two

different limits:

Limit of a function at a point:

In many cases, when the limit of the function f at the point c is well-defined, and when c isactually in the domain of definition of the function f(x) then


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3.3. LIMITS AT A POINT 75

Example:

2 1.5 1 0.5 0 0.5 1 1.5 21.5

1

0.5

0

0.5

1

1.5

x

f(x) = sin(x)

However, there are exceptions...

3.3.2 Gaps, exclusions and asymptotes

Gaps:

Example:

4 3 2 1 0 1 2 3 43

2

1

0

1

2

3

x

f(x) = x1 if x < 0

f(x) = x if x > 0

Asymptotes:


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Example:

4 3 2 1 0 1 2 3 420

15

10

5

0

5

10

15

20

x

f(x)= 1/x

Exclusions:

Example:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

9

10

x

f(x) = (x29)/(x3)


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3.3.3 Limit laws and behavior tables

Limit laws and behavior tables are exactly the same as in the case of limits at infinity.

3.3.4 Finding limits at a point - a multi-step process, and lots of

examples

Finding limits at a point is usually easier than finding limits at infinity, but not always. Hereare some guidelines as to how to proceed:

Identify the point at which the limit is considered. Is the function defined in a different way on the left and on the right of the point? If

so, there may be a gap. You need to limit from the left and from the right to establishwhether there is a gap or not.Examples:

Let f(x) = sin(x) when x 2 and f(x) = cos(x) when x > 2. What islimx2 f(x)?

Let f(x) = sin(x) when x 1/4 and f(x) = cos(x) when x > 1/4. What islimx1/4 f(x)?

What is the domain of definition of the function? If the function has no gap, and c isnot outside, or on the edge of the domain of definition, then

Examples:

Let f(x) = e3x. What is limx2 f(x)?

Let f(x) = 1/x. What is limx1 f(x)?


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If c is on the edge of the domain of definition, one of the two limits (from the left orfrom the right) is not defined. The other one needs to be studied carefully.Examples:

Let f(x) =

x2 1. What is limx1 f(x)?

Let f(x) = 12x2 . What is limx

2 f(x)?

Ifc is a single point that is excluded from the domain of definition, then the limit mayexist or not, depending on whether the function has an asymptote or an exclusion. Ifthere is an asymptote, it must be studied using a signs table. Exclusions can come indifferent varieties, and must be studied carefullyExamples:

Let f(x) = 1x21 . What is limx1 f(x)?

Let f(x) = x22x+1x1 . What is limx1 f(x)?


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Finally, some cases may be well-known trigonometric limits, or power/logarithmic limitwhich follow the rules listed in the next Sections.

3.3.5 Limits of powers and logarithms at x = 0

Rule:

so that

Examples:

limx0 x ln(x):

limx0 x1/2 log2(x):

limx0 x1 1ln(x)

3.3.6 Sandwich Theorem at a point

The Sandwich Theorem at a point is very similar to the one at infinity:Rule:

Examples:


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limx0 x sin(x):

limx0 x sin1x

:


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Check your understanding of Lecture 11

Sketching functions and their limitsFor each of the following, give an example of a function f(x) that satisfies the require-

ment. Write down the expression for the function, and plot the function on a graphwith suitable scales

A function that has an asymptote at x = 10.

A function that satisfies limx2 = 4.

A function that satisfies limx0 = 2 and limx0+ = 1. A function that satisfies limx0 = + and limx0+ = 0.

Limits at a point. Textook pages 127-128Problems 1, 3, 5, 10, 12, 21, 22, 23, 26, 27.

Limits of powers/logarithms: Find the limit (or the behavior, when appropriate)of the following functions

limx0+ x1/3 ln(x)

limx0+ exx ln(x)

limx0+x

ln(x)

Sandwich Theorem Textbook page 148Problem 1

Limits at a Point

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