   Chapter 1, Problem 45RE

Chapter
Section
Textbook Problem

# Let f ( x ) = { − x if   x < 0 3 − x if   0 ≤ x < 3 ( x − 3 ) 2 if   x > 3 (a) Evaluate each limit, if it exists. (i) lim x → 0 + f ( x ) (ii) lim x → 0 − f ( x ) (iii) lim x → 0 f ( x ) (iv) lim x → 3 − f ( x ) (v) lim x → 3 + f ( x ) (vi) lim x → 3 f ( x ) (b) Where is f discontinuous? (c) Sketch the graph of f.

(a)

To determine

To evaluate: The limit, if it exists.

Explanation

Given:

The function, f(x)={xif x<03xif 0x<3(x3)2if x>3.

That is, if x<0, then f(x)=x, if 0x<3, then f(x)=3x and if x>3, then f(x)=(x3)2.

Theorem used:

1. The functions such as “Polynomials, rational functions, root functions, trigonometric functions, inverse trigonometric functions, exponential functions and logarithmic functions” are continuous at every number in their domains.

2. A function f is continuous from the right at a number a if limxa+f(x)=f(a) and a function f is continuous from the left at a number a if limxaf(x)=f(a).

3. The limit limxaf(x)=L if and only if limxaf(x)=L=limxa+f(x).

Section (i)

Obtain the limit of the function f(x) as x approaches right side of 0.

If 0x<3, then f(x)=3x.

Here, f(x)=3x is a polynomial defined in the interval [0,3) and by theorem 1, it is continuous everywhere on its domain [0,3). That is, limxaf(x)=f(a) for every a[0,3). Therefore, the limit exist.

Consider the right hand limit limx0+f(x).

limx0+f(x)=limx0+(3x)=f(0)[by theorem 2]=3(0)=3

Thus, the limit of the function f(x)=3x as x approaches right side of 0 is 3.

Section (ii)

Obtain the limit of the function f(x) as x approaches left side of 0.

If x<0, then f(x)=x.

Here, f(x)=x is a root function defined in the interval (,0) and by theorem 1, it is continuous everywhere on its domain (,0). That is, limxaf(x)=f(a) for every a(,0). Therefore, the limit exist.

Consider the left hand limit limx0f(x).

limx0f(x)=limx0(x)=0[by theorem 2]=0

Thus, the limit of the function f(x)=x as x approaches left side of 0 is 0

(b)

To determine

To find: The function f is discontinuous at which numbers.

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