Question
Asked Oct 14, 2019
7
Exercise 5: Calculate the following limits, using the 6-e definition of a limit (i.e., provide an e-8 proof for the
following limits)
(a) lim |1 - 3x|
()
+2
(b) lim 2 - 5x
ft auU :
Pd
(c) lim r3
Ix 2
(d) lim f(x) where
(h
Ahoe)*
f (x) =
H 3
,2
x < 3
7,
2x3, x >3
x 3
dalw
boowloong u
t
help_outline

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7 Exercise 5: Calculate the following limits, using the 6-e definition of a limit (i.e., provide an e-8 proof for the following limits) (a) lim |1 - 3x| () +2 (b) lim 2 - 5x ft auU : Pd (c) lim r3 Ix 2 (d) lim f(x) where (h Ahoe)* f (x) = H 3 ,2 x < 3 7, 2x3, x >3 x 3 dalw boowloong u t

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Step 1

Hey, since there are multiple questions posted, we will answer first three sub parts question. If you want any specific question to be answered, then please submit that question only or specify the question number in your message.

Step 2
Definition:
Let fbe a real valued function defined of a subset I of the real numbers. Let c be a limit
point of I and let L be real number. We say that lim f(x) L if every > 0 there exist
a positive number 0 such thatf(x)-L<E whenever 0<r-c<
for all
(a)
Consider the given limit as lim1-3x|
x+2
By guess the limit of the limit is lim 1-3x5
x2
Now, the limit can be prove using (E,8) as follows.
f(x)-L-3x-5| (f(x)1-3x and L -5)
--4-3x
=3x + 4
3x-6+10
5(x-2)+10
<3x-2+10
E 10
Choose 5
3
help_outline

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Definition: Let fbe a real valued function defined of a subset I of the real numbers. Let c be a limit point of I and let L be real number. We say that lim f(x) L if every > 0 there exist a positive number 0 such thatf(x)-L<E whenever 0<r-c< for all (a) Consider the given limit as lim1-3x| x+2 By guess the limit of the limit is lim 1-3x5 x2 Now, the limit can be prove using (E,8) as follows. f(x)-L-3x-5| (f(x)1-3x and L -5) --4-3x =3x + 4 3x-6+10 5(x-2)+10 <3x-2+10 E 10 Choose 5 3

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Step 3
(b)
Consider the given limit as lim (2- 5x)
x-0
By guess the limit of the limit is lim(2-5.x)= 2.
x0
Now, the limit can be prove using (e,8) as follows
f(x)-L 2-5x-2
=5x
5x
5x- 0
f(x)2-5x and L 2)
Choose S
<&
help_outline

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(b) Consider the given limit as lim (2- 5x) x-0 By guess the limit of the limit is lim(2-5.x)= 2. x0 Now, the limit can be prove using (e,8) as follows f(x)-L 2-5x-2 =5x 5x 5x- 0 f(x)2-5x and L 2) Choose S <&

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Math

Calculus

Limits