BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2, Problem 8RE
To determine

To find: The limit of the function limt2t24t38.

Expert Solution

Answer to Problem 8RE

The limit of the function is 13_.

Explanation of Solution

Formula used:

Direct substitution property:

If f is a polynomial or a rational function and a is in the domain of f, then limxaf(x)=f(a).

Difference of cubes formula: (a3b3)=(ab)(a2+ab+b2)

Difference of squares formula: (a2b2)=(a+b)(ab)

Fact 1:

If f(x)=g(x) when xa, then limxaf(x)=limxag(x), provided the limit exist.

Given:

Let f(t)=t24t38 (1)

Note 1:

The direct substitution method is not applicable for the function f(t) as the function f(1) is in an indeterminate form at t=1.

f(2)=(2)24(2)38=4488=00

Note 2:

The limit may be infinite or it may be some finite value when both the numerator and the denominator approach 0.”

Calculation:

By note 2, consider the limit t approaches 2 but t2.

Simplify f(t) by using elementary algebra.

f(t)=t24t38=t2(2)2t3(2)3

Apply the difference of squares formula in the denominator,

f(t)=(t+2)(t2)t3(2)3

Apply the difference of cubes formula in the numerator,

f(t)=(t+2)(t2)(t2)(t2+2t+22)=(t+2)(t2)(t2)(t2+2t+4)

Since the limit of t approaches to 2 but not equal to 2, cancel the common term t20 from both the numerator and the denominator,

f(t)=(t+2)(t2+2t+4)

By fact 1, if f(t)=(t+2)(t2+2t+4) and t2, then limt2t24t38=limt2((t+2)(t2+2t+4)).

Apply the direct substitution property on the limit function.

limt2((t+2)(t2+2t+4))=(2+2)(22+2(2)+4)=4(4+4+4)=412=13

Thus, the limit of the function is 13_.

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