   Chapter 2, Problem 1RE

Chapter
Section
Textbook Problem

The graph of f is given. (a) Find each limit, or explain why it does not exist.(i) lim x → 2 + f ( x ) (ii) lim x → − 3 + f ( x ) (iii) lim x → − 3 f ( x ) (iv) lim x → 4 f ( x ) (v) lim x → 0 f ( x ) (vi) lim x → 2 − f ( x ) (vii) lim x → ∞ f ( x ) (viii) lim x → − ∞ f ( x ) (b) State the equation of the horizontal asymptotes.(c) State the equations of the vertical asymptotes.(d) At what numbers is f discontinuous? Explain.

(a)

To determine

To find: The limit of each of the given functions and explain if the limit does not exist.

Explanation

Calculation:

Section (i)

Obtain the value of limx2+f(x).

From the given graph, it is found that the curve move towards y=3 as x approaches 2 from the right, that is x>2.

Therefore, limx2+f(x)=3.

Section (ii)

Obtain the value of limx3+f(x).

From the given graph, it is found that the curve move towards y=0 as x approaches 3 from the right, that is x>3.

Therefore, limx3+f(x)=0.

Section (iii)

Obtain the value of limx3f(x).

From section (ii),

limx3+f(x)=0

From the given graph, it is found that the curve move towards y=2 as x approaches 3 from the left, that is x<3.

Thus, limx3+f(x)=2

Recall the definition that limxaf(x) exists only when limxaf(x)=limxa+f(x).

Since limx3+f(x)limx3f(x)

Therefore, limx3f(x) does not exists

(b)

To determine

To state: The equation of horizontal asymptotes.

(c)

To determine

To state: The equation of vertical asymptotes.

(d)

To determine

To find: The numbers at which f is discontinuous.

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