# Whether the statement “If f ′ ( r ) exists then lim x → r f ( x ) = f ( r ) ” is true.

### Single Variable Calculus: Concepts...

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

### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter 2, Problem 16RQ
To determine

## To find: Whether the statement “If f′(r) exists then limx→rf(x)=f(r) ” is true.

Expert Solution

The statement “I If f(r) exists then limxrf(x)=f(r) ” is true

### Explanation of Solution

Given information:

The given statement is “ f(r) exists”

Calculation:

Since f(r) exists, consider xr .

f(x)f(r)=f(x)f(r)xr(xr))

Take limit both the sides.

limxrf(x)f(r)=limxrf(x)f(r)xr×limxr(xr))limxrf(x)f(r)=f(r)(rr)limxrf(x)f(r)=0

Further solve.

limxrf(x)=limxrf(r)=f(r)

Therefore, the statement “If f(r) exists then limxrf(x)=f(r) ” is true

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