## Calculus AB and Calculus BC

## CHAPTER 2 Limits and Continuity

**Concepts and Skills**

In this chapter, you will review

• general properties of limits;

• how to find limits using algebraic expressions, tables, and graphs;

• horizontal and vertical asymptotes;

• continuity;

• removable, jump, and infinite discontinuities;

• and some important theorems, including the Squeeze Theorem, the Extreme Value Theorem, and the Intermediate Value Theorem.

### A. DEFINITIONS AND EXAMPLES

The number *L* is the *limit of the function f* (*x*) as *x* approaches *c* if, as the values of *x* get arbitrarily close (but not equal) to *c*, the values of *f* (*x*) approach (or equal) *L*. We write

In order for to exist, the values of *f* must tend to the same number *L* as *x* approaches *c* from either the left or the right. We write

**One-sided limits**

for the *left-hand limit* of *f* at *c* (as *x* approaches *c* through values *less* than *c*), and

for the *right-hand limit* of *f* at *c* (as *x* approaches *c* through values *greater* than *c*).

**EXAMPLE 1**

The greatest-integer function *g*(*x*) = [*x*], shown in Figure N2–1, has different left-hand and right-hand limits at *every* integer. For example,

This function, therefore, does not have a limit at *x* = 1 or, by the same reasoning, at any other integer.

**FIGURE N2–1**

However, [*x*] does have a limit at every nonintegral real number. For example,

**EXAMPLE 2**

Suppose the function *y* = *f* (*x*), graphed in Figure N2–2, is defined as follows:

Determine whether limits of *f*, if any, exist at

(a) *x* = −2,

(b) *x* = 0,

(c) *x* = 2,

(d) *x* = 4.

**FIGURE N2–2**

**SOLUTIONS:**

**(a)** so the right-hand limit exists at *x* = −2, even though *f* is not defined at *x* = −2.

**(b)** does not exist. Although *f* is defined at *x* = 0 (*f* (0) = 2), we observe that whereas For the limit to exist at a point, the left-hand and right-hand limits must be the same.

**(c)** This limit exists because Indeed, the limit exists at *x* = 2 even though it is different from the value of *f* at 2 (*f* (2) = 0).

**(d)** so the left-hand limit exists at *x* = 4.

**EXAMPLE 3**

Prove that

**SOLUTION:** The graph of |*x*| is shown in Figure N2–3.

We examine both left- and right-hand limits of the absolute-value function as *x* → 0. Since

it follows that

Since the left-hand and right-hand limits both equal 0,

Note that if *c* > 0 but equals −*c* if *c* < 0.

**FIGURE N2–3**

DEFINITION

The function *f* (*x*) is said to *become infinite* (positively or negatively) as *x* approaches *c* if *f* (*x*) can be made arbitrarily large (positively or negatively) by taking *x* sufficiently close to *c.* We write

Since for the limit to exist it must be a finite number, neither of the preceding limits exists.

This definition can be extended to include *x* approaching *c* from the left or from the right. The following examples illustrate these definitions.

**EXAMPLE 4**

Describe the behavior of near *x* = 0 using limits.

**SOLUTION:** The graph (Figure N2–4) shows that:

**FIGURE N2–4**

**EXAMPLE 5**

Describe the behavior of near *x* = 1 using limits.

**SOLUTION:** The graph (Figure N2–5) shows that:

**FIGURE N2–5**

Remember that none of the limits in Examples 4 and 5 exists!

DEFINITION

We write

if the difference between *f* (*x*) and *L* can be made arbitrarily small by making *x* sufficiently large positively (or negatively).

In Examples 4 and 5, note that

**EXAMPLE 6**

From the graph of (Figure N2–6), describe the behavior of *h* using limits.

**SOLUTION:**

**FIGURE N2–6**

DEFINITION

The theorems that follow in §C of this chapter confirm the conjectures made about limits of functions from their graphs.

Finally, if the function *f* (*x*) becomes infinite as *x* becomes infinite, then one or more of the following may hold:

END BEHAVIOR OF POLYNOMIALS

Every polynomial whose degree is greater than or equal to 1 becomes infinite as *x* does. It becomes positively or negatively infinite, depending only on the sign of the leading coefficient and the degree of the polynomial.

**EXAMPLE 7**

For each function given below, describe

(a) *f* (*x*) = *x*^{3} − 3*x*^{2} + 7*x* + 2

**SOLUTION:**

(b) *g*(*x*) = −4*x*^{4} + 1,000,000*x*^{3} + 100

**SOLUTION:**

(c) *h*(*x*) = −5*x*^{3} + 3*x*^{2} −4π + 8

**SOLUTION:**

(d) *k*(*x*) = π − 0.001*x*

**SOLUTION:**

It’s easy to write rules for the behavior of a polynomial as *x* becomes infinite!