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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f.

f ( x ) = { x + 2 if x < 0 e x if 0 x 1 2 x if x > 1

To determine

To find: The function f(x)={x+2if x<0exif 0x12xif x>1 is discontinuous at which numbers and explain for which of the numbers are continuous from the right, from the left, or neither. Sketch the graph of the function f(x)={x+2if x<0exif 0x12xif x>1.

Explanation

Definition used: “A function f is continuous at a number a if limxaf(x)=f(a)”.

Note 1: “If f is defined near a, f is discontinuous at a whenever f is not continuous at a”.

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).

4. If f is continuous at b and limxag(x)=b, then limxaf(g(x))=f(limxag(x)).

Calculation:

By note 1, the function f is said to be discontinuous at x=a if anyone of the following conditions does not satisfied.

  • f(a) is defined
  • The limit of the function at the number a exists.
  • limxaf(x)=f(a)

Consider the piecewise function f(x)={x+2if x<0exif 0x12xif x>1 .

Here, the function f(x)=x+2 is a polynomial function defined in the interval (,0), f(x)=ex is a exponential function defined in the interval (0,1) and f(x)=2x is a polynomial function defined in the interval (1,).

Since f(x)=x+2 is a polynomial function, f(x)=ex is an exponential function and f(x)=2x is a polynomial function and by theorem 1, those functions are continuous on its respective domains.

Therefore, f is continuous on the interval (,0)(0,1)(1,).

So that, f might be discontinuous at 0 and 1.

Check the discontinuity of f at x=0.

At x=0, then f(0)=1 is defined. (1)

The limit of the function f(x) as x approaches a=0 is computed as follows.

Consider the left hand limit limx0f(x)

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