Q. 9 If聽limx鈫抍fx-fcx-c exists, wha... [FREE SOLUTION]

Chapter 2: Q. 9 (page 183)

If $\lim_{x 鈫� c} \frac{f (x) - f (c)}{x - c}$ exists, what can you say about the differentiability of $f$ at $x = c$ ? What can you say about the continuity of $f$ at $x = c$ ?

Short Answer

Expert verified

The function is differentiable as well as continuous at $x = c$ .

Step by step solution

Step 1. Given information

It is given that $\lim_{x 鈫� c} \frac{f (x) - f (c)}{x - c}$ exists, we have to say about the differentiability and continuity of $f$ at $x = c$ .

Step 2. Check differentiability and continuity

Since $\lim_{x 鈫� c} \frac{f (x) - f (c)}{x - c}$ is exist, therefore

$\lim_{x 鈫� c^{-}} \frac{f (x) - f (c)}{x - c} = \lim_{x 鈫� c^{+}} \frac{f (x) - f (c)}{x - c}$

Thus the function is differentiable at $x = c$ .

The function is differentiable at $x = c$ only when it is continuous at $x = c$ .

Hence the function is continuous at $x = c$ .

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91影视

If $\lim_{x 鈫� c} \frac{f (x) - f (c)}{x - c}$ exists, what can you say about the differentiability of $f$ at $x = c$ ? What can you say about the continuity of $f$ at $x = c$ ?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Check differentiability and continuity

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91影视

Iflimx鈫�cfx-fcx-c exists, what can you say about the differentiability of fat x=c? What can you say about the continuity of fat x=c?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Check differentiability and continuity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

If $\lim_{x 鈫� c} \frac{f (x) - f (c)}{x - c}$ exists, what can you say about the differentiability of $f$ at $x = c$ ? What can you say about the continuity of $f$ at $x = c$ ?