Module 5 - Limits and Infinity

Module 5 - Answers

Lesson 1

Answer 1

5.1.1

It appears that:

Vertical asymptote: x = 3
Horizontal asymptote: y = 2

You cannot really be sure from the graph. You need to apply analytic techniques as we will illustrate.

Answer 2

5.1.2 As x approaches 3 from the left, the function values are negative and their absolute values increase without bound. Symbolically, as x

3^–, f(x)

–

. As x approaches 3 from the right, the function values are positive and increase without bound. Symbolically, as, x

3⁺, f(x)

Answer 3

5.1.3

Because the values of the function in absolute value increase without bound as x approaches 3 from the left and from the right, x = 3 is a vertical asymptote. Notice that

means as x 3^–, f(x) –

means as, x 3⁺, f(x)

Answer 4

5.1.4 The tables show that as the magnitude (absolute value) of the x-coordinates increases, the y-coordinates approach 2. Symbolically, f(x)

2 as x

–

and as x

Lesson 2

Answer 1

5.2.1

Because , which is not or – , x = –2 is not a vertical asymptote.

Answer 2

5.2.2

Because , there is a hole at x = –3 and not a vertical asymptote there.

Because and , x = 2 is a vertical asymptote.

The Window values shown are [–7.9, 7.9] x [–10, 10] with xres = 1.

There is a tiny hole at and x = 2 is a vertical asymptote. Because xres is 1 and a pixel represents x = 2, the calculator evaluated the function at x = 2 and determined that the function was not defined there. That is why there is no vertical line shown at x = 2.

Lesson 3

Answer 1

5.3.1

Because the function does not approach a finite real number as the magnitude of x gets large without bound, no horizontal line is an asymptote.

Answer 2

5.3.2

The vertical asymptote is x = 2 and the oblique asymptote is y = x + 2. In a large window the graph of is similar to y = x + 2, the oblique asymptote. In a window near x = 2 the graph is similar to .

The Window shown is [–30, 30] x [–30, 30], xscl = yscl = 5, and the oblique asymptote y = x + 2 is shown as a dotted line.

The style of a selected graph may be chosen from the Style menu, which is displayed by pressing in the Y= editor while the function in question is highlighted. The style of y = x + 2 was chosen to be dotted.