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

Question

Part d please

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

Left hand limit of x=0 is=0+1=1

Right hand limit of x=0 is= -infinity 

So, at x=0 it is discontinuous

limh(x)
x-0
= lim (x+1)
X0
-0+1
lim h(x)
x-0
lim In x
x0
=-oo
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limh(x) x-0 = lim (x+1) X0 -0+1 lim h(x) x-0 lim In x x0 =-oo

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Step 2

Left hand limit of x=1 is =0

Right hand limit of x=1 is = 1

So h(x) is discontinuous at x=1

Answer(a): h(x) is discontinuous at x=0 and x=1

limh(x)
x1
limln(x)
x-1
=In(1)
=0
lim h(x)
x1
=lm (2-x)
x1
-2-1
=1
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limh(x) x1 limln(x) x-1 =In(1) =0 lim h(x) x1 =lm (2-x) x1 -2-1 =1

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Step 3

Both at x=0 and x=1 it is left continuous , because equality holds at left...

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Tagged in

Math

Calculus

Continuity

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