Question 2. Let f(x) = 2x - 5. It is given that lim f (x) = 1. x-3 Complete the following proof using the ɛ and & definition of the limit: Proof: Let ɛ > 0 be an arbitrary real number. It is needed to find a real number 8 > 0 such that, if |x – 3| < 8 then If (x) – 1| < ɛ. Now, If (x) – 1| < ɛ if and only if -ɛ < f(x) – 1 < E if and only if if and only if if and only if if and only if |x – 3| < .... Now, for every ɛ > 0, we can choose & which implies that, " if |x - 3| < then If (x) 1|<..." ...... is satisfied. Hence lim f(x) = 1 x-3 Proof is completed.
Question 2. Let f(x) = 2x - 5. It is given that lim f (x) = 1. x-3 Complete the following proof using the ɛ and & definition of the limit: Proof: Let ɛ > 0 be an arbitrary real number. It is needed to find a real number 8 > 0 such that, if |x – 3| < 8 then If (x) – 1| < ɛ. Now, If (x) – 1| < ɛ if and only if -ɛ < f(x) – 1 < E if and only if if and only if if and only if if and only if |x – 3| < .... Now, for every ɛ > 0, we can choose & which implies that, " if |x - 3| < then If (x) 1|<..." ...... is satisfied. Hence lim f(x) = 1 x-3 Proof is completed.
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Transcribed Image Text:Question 2.
Let f(x) = 2x – 5. It is given that lim f(x) = 1.
x-3
Complete the following proof using the ɛ and & definition of the limit:
Proof: Let ɛ > 0 be an arbitrary real number. It is needed to find a real number 8 > 0 such that,
if |x – 3| < 8 then If (x) – 1| < E.
Now, If (x) – 1| < ɛ
if and only if
-ɛ < f(x) – 1 < E
if and only if
if and only if
if and only if
if and only if |x – 3| < ...
Now, for every ɛ > 0, we can choose 8
which implies that, " if x - 3| <
then If (x) 1|<..."
......
is satisfied. Hence lim f(x) = 1
x-3
Proof is completed.
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