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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

Use continuity to evaluate the limit.

lim x 2 x 20 x 2

To determine

To evaluate: The limit of the function f(x)=x20x2 as x approaches 2 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:

4. If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:

(i) f+g (ii) fg (iii) cf (iv) fg (v) fg if g(a)0.

5. (a) Any polynomial is continuous everywhere; that is, it is continuous on =(,).

(b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

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.

Calculation:

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

Consider the function f(x) is product of two functions. That is, f(x)=p(x)q(x) where p(x)=x and q(x)=20x2.

The function p(x)=x is defined for every real numbers.

The function q(x)=20x2 is defined whenever 20x2 must be greater than or equal to zero.

20x2020x22020x220

Multiply both sides of the inequality by −1,

x220x20 and x2020x20

This implies that, the function q(x) is defined on the interval [20,20].

Therefore, the domain of the function f(x) is D=[20,20]

Thus, the domain of the function f(x) is D=[20,20]

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