#### To determine

**To state:**

Whether the given statement,"If f’(r) exists, then limx→rfx=f(r)" is true or false.

#### Answer

True.

#### Explanation

**Given:** f'(r) exists.

From definition

f'r=limh→0fr+h-frh

f'(r) exists.

limh→0fr+h-frh exists.

Put x=r+h

As h→0, x→r

f'r=limx→rfx-frx-r

Separating limit,

f'r=limx→rfx-frlimx→r(x-r)

Multiply by limx→r(x-r) on both side,

f'rlimx→rx-r=limx→rfx-fr

f'r.0=limx→rfx-fr

0=limx→rfx-fr

Separating limit,

limx→rfx-limx→rf(r)=0

limx→rfx-f(r)limx→r1=0

limx→rfx=f(r)

The statement limx→rfx=f(r) essentially means that f is continuous at x = r. Existence of f’(r) means than f is differentiable at r. So this statement is equivalent to the statement that if f is differentiable at x =r then it is continuous at r.

**Final statement:**

If f’(r) exists, then limx→rfx=f(r) is true statement.