 # Limits and continuity, differentiation, integration, homework help Anonymous
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6 questions

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MrClyneHelp
School: UC Berkeley   there you go, all questions are explained and I believe the question is fully answered

Surname 1

Name
Professor
Course
Date

1. Determine whether the following functions are continuous
In order to determine if a function is continuous at a point 𝑥 = 𝑎 , three conditions need
to be satisfied.
i.

the function f is defined at a,

ii.

the limit of f as x approaches a from the right-hand and left-hand limits exist and are
equal,

iii.

the limit of f as x approaches a is equal to f(a).

If any of the conditions is not met the function is not continuous.

a. 𝑓(𝑥) = {

𝑥 − 2, 𝑥 < 1
𝑥≥1
√𝑥,

When 𝑥 = 1, 𝑓(𝑥) = 𝑥 − 2 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 1 − 2 = −1 𝑎𝑛𝑑 𝑓(𝑥) = √𝑥 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 √1 = 1
The solutions are not equal hence the function is not continuous.
1

𝑥 𝑓𝑜𝑟 𝑥 < 2
0 𝑓𝑜𝑟

b. 𝑓(𝑥) =

1

𝑥=2
1

{

1 − 𝑥 𝑓𝑜𝑟 𝑥 > 2

Surname 2

1

1

1

1

1

𝑤ℎ𝑒𝑛 𝑥 < 2 , 𝑓(𝑥) = 2 𝑤ℎ𝑒𝑛 𝑥 = 2 , 𝑓(𝑥) = 0 𝑎𝑛𝑑 𝑤ℎ𝑒𝑛 𝑥 > 2 , 𝑓(𝑥) = 2 hence the function
is not continuous.

2. Find the differential of the following
a. 𝑓(𝑥) = 100𝑥 4 + 8𝑥 3 − 𝑥 2 + 24𝑥 − 8
In this case, bring the power of x down and multiply by the coefficient of x then reduce the
power by one and sinc...

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Excellent job Brown University

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