   Chapter 2.3, Problem 51E

Chapter
Section
Textbook Problem

Let B ( t ) = { 4 − 1 2 t   if   t < 2 t + c   if   t ≥ 2 Find the value of c so that lim t → 2 B ( t ) exists.

To determine

To find: The value of c when the limit exists.

Explanation

Given:

The given function is B(t)={412tif t<2t+cif t2.

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).

Theorem 1:

The limit limxaf(x)=L if and only if limxaf(x)=L=limxa+f(x).

Obtain the value of c when the limit exists.

By theorem 1, the limit exists if and only if the left-hand and right-hand side limits are equal.

Obtain the limit of the function B(t) as t approaches left-hand side of 2.

Since B(t)=412t for t<2,

limt2(B(t))=limt2(412t)=412(2) [by direct substitution]=41=3

Thus, the limit of the function B(t) as t approaches left-hand side of 2 is 3

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