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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

Locate the discontinuities of the function and illustrate by graphing.

y = 1 1 + e 1 / x

To determine

To locate: The discontinuities of the function y=11+e1x and sketch the graph of the function y=11+e1x.

Explanation

The domain is the set of all input values of the function for which the function is real and defined.

Consider the denominator of the function y(x) and equate to zero to obtain the undefined points.

Since the denominator of y(x) is 1+e1x, the undefined points are obtained as shown below.

1+e1x=0e1x=1

Since e1x cannot be zero or negative for x, there is no undefined points in .

But, for x=0, the function y=11+e1x is undefined.

Therefore, the domain of the function is the set of all real numbers except zero.

And the interval notation of the domain of y(x) is (,0)(0,).

Suppose the function y=f(x) is of the form f(x)=p(x)q(x) where p(x)=1 and q(x)=1+e1x.

Here, p(x) is a constant function and it is continuous everywhere.

The function q(x)=1+e1x is a combination of a constant function and the composite of exponential and rational function.

The rational function 1x is continuous on its domain (,0)(0,). The exponential function is continuous everywhere. Therefore, the composite function e1x is continuous on its domain (,0)(0,).

This implies that, q(x) is continuous on the interval (,0)(0,).

Thus, f(x) is continuous on the interval (,0)(0,).

It is clear that, f(x) is continuous for every real numbers except x=0.

Substitute x=0 in f(x),

f(0)=11+e10

Thus, the function f(0) is undefined.

The limit of the function f(x) as x approaches zero is computed as follows.

Consider the left hand limit limx011+e1x

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