   Chapter 2.5, Problem 38E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use continuity to evaluate the limit. lim x → 4 3 x 2 − 2 x − 4

To determine

To evaluate: The limit of the function f(x)=3x22x4 as x approaches 4 by using continuity and check the function is continuous or not.

Explanation

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

Theorems used:

7. 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.

9. If g is continuous at a and f is continuous at g(a), then the composite function fg given by (fg)(x)=f(g(x)) is continuous at a.

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

Calculation:

The function f(x)=3x22x4 is defined whenever x22x4 must be greater than are equal to zero.

x22x40(x(1+5))(x(15))0x15 or x1+5

Thus, the domain of the function f(x) is D=(,15][1+5,).

Consider the function f(x) is composition of two functions. That is, f(x)=h(g(x)) where h(x)=3x and g(x)=x22x4.

The function g(x)=x22x4 is a composition of root and polynomial functions. That is, g(x)=r(s(x)) where r(x)=x and s(x)=x22x4

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