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Problem 3 - Let g(x) and h(x) be the following piece-wise defined functionsx1, if x

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Problem 3 in attached image

Problem 3 - Let g(x) and h(x) be the following piece-wise defined functions
x1, if x<0
if 0 1
g(ax)
IC
2 , if 1 <x
1, if x < 0
h(ar) =
if 0 x 1
In x,
2 x, if 1 << x
a) Find the points where g(x) is discontinuous
b) At which of these points is g(x) right-continuous, left-continuous or neither?
c) State the type of discontinuity at these points (removable, jump, or infinite)
d) Repeat a), b), c) for h(x)
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Problem 3 - Let g(x) and h(x) be the following piece-wise defined functions x1, if x<0 if 0 1 g(ax) IC 2 , if 1 <x 1, if x < 0 h(ar) = if 0 x 1 In x, 2 x, if 1 << x a) Find the points where g(x) is discontinuous b) At which of these points is g(x) right-continuous, left-continuous or neither? c) State the type of discontinuity at these points (removable, jump, or infinite) d) Repeat a), b), c) for h(x)

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Step 1

Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and specify the other subparts (up to 3) you’d like answered.

Possible discontinuities of g(x) are at x=0 and x=1.

We find left and right hand limits at x=0 and x=1.

 

Step 2

At x=0, left hand limit=0+1=1

And right hand limit = e^0=1

So at x=0, g(x) is continuous. 

At x=1 left hand limit=e^1=e

Right hand limit=2-1=1

So, at x=1 the graph is discontinuous. 

Answer(a): x=1

3
2
-3
-2
2
3
-1
-2
LO
C
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3 2 -3 -2 2 3 -1 -2 LO C

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Step 3

g(1)= left hand of g(x) limit at x=1

Answer(b)...

g(1)= lim g(x)1
x1
2
-3
2
C
2
4
--1
-2
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g(1)= lim g(x)1 x1 2 -3 2 C 2 4 --1 -2

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Tagged in

Math

Calculus

Continuity

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