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Chapter 2: Q. 8 (page 183)

The function fx=4-x2is both continuous and differentiable at x = 1. Write these facts as limit statements.

Short Answer

Expert verified

The function is continuous and differentiable atx=1.

Step by step solution

01

Step 1. Given information

We have to prove that the function fx=4-x2is both continuous and differentiable at x=1.

02

Step 2. Check the function is continuous

Left hand limit,

limx鈫�1-4-x2=4-12=3

Right hand limit,

limx鈫�1+4-x2=4-12=3

At point x=1, the value of function is

f1=4-12=3

Therefore, the function is continuous atx=1.

03

Step 3. Check the function is differentiable.

Left hand derivative,

limh鈫�0-f1+h-f1h=limh鈫�0-4-1+h2-4-1h=limh鈫�0--2h-h2h=limh鈫�0--2-h=-2

Right hand derivative,

limh鈫�0+f1+h-f1h=limh鈫�0+4-1+h2-4-1h=limh鈫�0+-2h-h2h=limh鈫�0+-2-h=-2

Therefore, the function is differentiable atx=1.

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