Chapter 2.6, Problem 64E

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8th Edition

James Stewart

ISBN: 9781305270343

Textbook Problem

Find the limits as *x → ∞* and as *x → –∞*. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

**Example 12**

**FIGURE 13**

*y* = (*x* – 2)^{4} (*x* + 1)^{3} (*x* – 1)

*y* = *x*^{2}(*x*^{2} – 1)^{2}(*x* + 2)

To determine

**To find:** The limit of the function *x* approaches infinity and negative infinity and obtain the *x* and *y* intercepts and sketch the rough graph of the function.

Explanation

**Limit Laws used:** Suppose that *c* is a constant and the limits

Limit law 1:

Limit law 2:

Limit law 3:

**Theorem used:**

**1.** If

**2.** If

**Note:**

The limit law

**Calculation:**

Obtain the *x* and *y* intercepts.

Consider the function,

The *y*-intercept is computed as follows,

Plug

Thus, the *y*-intercept is 0.

The *x*-intercept is computed as follows,

Plug

Thus, the *x* intercepts are −2, −1, 0 and 1.

Compute the value of the function as *x* approaches infinity.

Take the limit of *x* approaches infinity.

Here, *x* approaches infinity. Thus, the product *x* approaches infinity. That is,

Thus, the function *x* approaches infinity

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