Exercise 11: Consider the following functions:x -12 11S9(x),h(x), otherwise.g(x)h(x) =f (x)x21(a) Show that limg1 f(x) does not exist.Remark: A good, convincing, intuitive explanation, possibly with a couple graphs to help visualize,will receive full points. However, being able to write a full proof would be good.(b) For which values of c ER does lim>c f(x) exist?

Question
Asked Oct 13, 2019
Exercise 11: Consider the following functions:
x -1
2 1
1
S9(x),
h(x), otherwise.
g(x)
h(x) =
f (x)
x21
(a) Show that limg1 f(x) does not exist.
Remark: A good, convincing, intuitive explanation, possibly with a couple graphs to help visualize,
will receive full points. However, being able to write a full proof would be good.
(b) For which values of c ER does lim>c f(x) exist?
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Exercise 11: Consider the following functions: x -1 2 1 1 S9(x), h(x), otherwise. g(x) h(x) = f (x) x21 (a) Show that limg1 f(x) does not exist. Remark: A good, convincing, intuitive explanation, possibly with a couple graphs to help visualize, will receive full points. However, being able to write a full proof would be good. (b) For which values of c ER does lim>c f(x) exist?

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Expert Answer

Step 1

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We need to find the limit of the f(x) at x = 1, where f(x) is

хеQ
x21
х — 1
f(x) =
, otherwise
x2 - 1
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хеQ x21 х — 1 f(x) = , otherwise x2 - 1

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Step 2

In the neighbourhood of x = 1, we can encounter infinite numbers of rational and irrational numbers. Thus, the...

х — 1
CASE 1: Whenx is irrational, f(x) = h(x)
х2 — 1
LHL= lim f(x)
x1
х — 1
lim
х-1- х2 — 1
х — 1
using a2-b2 = (a - b)(a + b)]
lim
-т (х — 1)(х + 1)
1
lim
х-1 (х + 1)
1
2
x +1
CASE 2: When x is rational, f(x) = g(x)
x21
LHL = lim f(x)
x 1
x +1
lim
x1 x2 1
=
11
1
1+1
=
N
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х — 1 CASE 1: Whenx is irrational, f(x) = h(x) х2 — 1 LHL= lim f(x) x1 х — 1 lim х-1- х2 — 1 х — 1 using a2-b2 = (a - b)(a + b)] lim -т (х — 1)(х + 1) 1 lim х-1 (х + 1) 1 2 x +1 CASE 2: When x is rational, f(x) = g(x) x21 LHL = lim f(x) x 1 x +1 lim x1 x2 1 = 11 1 1+1 = N

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Tagged in

Math

Calculus

Limits