   Chapter 2, Problem 8RE

Chapter
Section
Textbook Problem

Find the limit. lim t → 2 t 2 − 4 t 3 − 8

To determine

To find: The limit of the function limt2t24t38.

Explanation

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

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