Finding Limits Graphically and Numerically 2015 Limits Introduction Copyright © Cengage Learning. All rights reserved. 1.2

Estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist. Study and use the informal definition of limit. Objectives

Formal definition of a Limit: If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L. “The limit of f of x as x approaches c is L.” This limit is written as

Limits can be found in various ways: a)Graphically b)Numerically c)Algebraically Ex: Find the following limit:

An Introduction to Limits Ex: Find the following limit:

Looks like y=1

An Introduction to Limits Ex: Find the following limit:

Start by sketching a graph of the function For all values other than x = 1, you can use standard curve-sketching techniques. However, at x = 1, it is not clear what to expect. We can find this limit numerically: An Introduction to Limits

To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x-values–one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table. An Introduction to Limits

The graph of f is a parabola that has a gap at the point (1, 3), as shown in the Figure 1.5. Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3. Using limit notation, you can write An Introduction to Limits This is read as “the limit of f(x) as x approaches 1 is 3.” Figure 1.5

This discussion leads to an informal definition of a limit: A limit is the value (meaning y value) a function approaches as x approaches a particular value from the left and from the right.

Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

At x=1:left hand limit right hand limit value of the function does not exist because the left and right hand limits do not match! DNE

At x=2:left hand limit right hand limit value of the function because the left and right hand limits match

At x=3:left hand limit right hand limit value of the function because the left and right hand limits match

You Try– Estimating a Limit Numerically Use the table feature of your graphing calculator to evaluate the function at several points near x = 0 and use the results to estimate the limit:

Example 1 – Solution The table lists the values of f(x) for several x-values near 0.

Example 1 – Solution From the results shown in the table, you can estimate the limit to be 2. This limit is reinforced by the graph of f (see Figure 1.6.) cont’d Figure 1.6

Use a graphing utility to estimate the limit:

Find

Limits That Fail to Exist

Show that does not exist Non-existance Because the behavior differs from the right and from the left of zero, the limit DNE.

Discuss the existence of the limit: Solution: Using a graphical representation, you can see that x does not approach any number. Therefore, the limit does not exist.

Example 5: xx→0 sin 1/x11 1 DNE Make a table approaching 0 The graph oscillates, so the limit does not exist.

Fig. 1.10, p. 51

Limits That Fail to Exist - 3 Reasons

Properties of Limits

Limits Basics Examples

Properties of Limits

Using Properties of Limits

Find the following limits:

Properties of Limits

Compute the following limits

Let’s take a look at the last one What happened when we plugged in 1 for x? When we get we have what’s called an indeterminate form Let’s see how we can solve it

Let’s look at the graph of Is the function continuous at x = 1?

You Try: Find the limit:

You Try:

Find the limit: Solution: By direct substitution, you obtain the indeterminate form 0/0. Example – Rationalizing Technique

In this case, you can rewrite the fraction by rationalizing the numerator. cont’d Solution

Now, using Theorem 1.7, you can evaluate the limit as shown. cont’d Solution

A table or a graph can reinforce your conclusion that the limit is. (See Figure 1.20.) Figure 1.20 Solution cont’d

Solution cont’d

Group Work : Sketch the graph of f. Identify the values of c for which exists.

Homework p all