Chapter 2: Q. 11 (page 183)

Suppose $f (1) = 3$ , $\lim_{x \to 1^{-}} f (x) = 3$ , $\lim_{x \to 1^{+}} f (x) = 3$ ,role="math" localid="1648483351627" $\lim_{x \to 0^{-}} \frac{f (1 + h) - f (1)}{h} = 2$ , $\lim_{x \to 0^{+}} \frac{f (1 + h) - f (1)}{h} = 0$
(a) Is $f$ continuous and/or differentiable at $x = 0$ ? What
about from the left or right?
(b) Sketch a possible graph of $f$ .

Short Answer

Expert verified

The function $f$ is continuous but not differentiable at $x = 1$ .

The possible graph can be as follow:

Step by step solution

Step 1. Given information

It is given that $f (1) = 3$ , $\lim_{x \to 1^{-}} f (x) = 3$ , $\lim_{x \to 1^{+}} f (x) = 3$ , $\lim_{h \to 0^{-}} \frac{f (1 + h) - f (1)}{h} = 2$ , $\lim_{h \to 0^{+}} \frac{f (1 + h) - f (1)}{h} = 0$

Step 2. Check continuity and differentiability at x=1

Since $f (1) = 3$ , $\lim_{x \to 1^{-}} f (x) = 3$ , $\lim_{x \to 1^{+}} f (x) = 3$ , therefore the function is continuous at $x = 1$ .

$\lim_{h \to 0^{-}} \frac{f (1 + h) - f (1)}{h} = 2$ and $\lim_{h \to 0^{+}} \frac{f (1 + h) - f (1)}{h} = 0$

Left hand derivatives and Right hand derivatives are not equal, So the function is not differentiable at $x = 1$ .

Step 3. Draw the possible graph of f.

The possible function will be

$f (x) = \{\begin{cases} 3, i f x \geq 0 \\ 2 x + 1, i f x < 0 \end{cases}$

So, the graph will be

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Check continuity and differentiability at x=1

Step 3. Draw the possible graph of f.

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Supposef1=3,limx→1-fx=3,limx→1+fx=3,role="math" localid="1648483351627" limx→0-f1+h-f1h=2,limx→0+f1+h-f1h=0(a) Is fcontinuous and/or differentiable at x=0? Whatabout from the left or right?(b) Sketch a possible graph of f.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Check continuity and differentiability at x=1

Step 3. Draw the possible graph of f.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Calculus

Applied Mathematics

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Decision Maths

Study anywhere. Anytime. Across all devices.