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Chapter 1: Q. 8 (page 119)

Given the following function f, define f(1)so that fis continuous at x=1, if possible:

f(x)=3x-1,ifx<1x2+1.ifx>1

Short Answer

Expert verified

As,f(1-)=f(1)=f(1+). Therefore,fis continuous atx=1.

Step by step solution

01

Step 1. Given information

Functionf(x)=3x-1,ifx<1x2+1,ifx>1

02

Step 2. Calculating f(1-)

Function is continuous if f(1-)=f(1)=f(1+).

f(1-)=limh鈫�0(3(1-h)-1)=limh鈫�0(3-3h-1)=3-3limh鈫�0h-1=3-0-1=2So,f(1-)=2

03

Step 3. Calculating f(1+)

f(1+)=limh鈫�0((1+h)2+1) =limh鈫�0(1+h2+2h+1)=1+limh鈫�0h2+2limh鈫�0h+1=1+0+0+1=2So,f(1+)=2

Therefore, the value of f(1)=2. Asf(1-)=f(1)=f(1+).

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Most popular questions from this chapter

Write a delta鈥揺psilon proof that proves that f is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.

f(x)=x鈭�1

Write delta-epsilon proofs for each of the limit statements limx鈫�cfx=Lin Exercises 47-60.

limx鈫�12x+4=6.

limx鈫�1ex-1e2x+2ex-3

Calculate each of the limits:

limh鈫�0(2+h)2-22h.

State what it means for a function f to be left continuous at a point x = c, in terms of the delta鈥揺psilon definition of limit.
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