Let . It is claimed that . Demonstrate the validity of this claim by completing the…

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

f(x)=5x-3

. It is claimed that

\displaystyle \lim_{x\to 2} f(x)=7

. Demonstrate the validity of this claim by completing the logical argument below. (a) Show that

|f(x)-7|

can be rewritten in the form

5|x-2|

. Provide the algebraic steps. (b) Suppose that

|x-2|<\delta

. Express

|f(x)-7|

entirely in terms of

\delta

. (c) Given an arbitrary

\varepsilon>0

, determine a value of

\delta

(as a function of

\varepsilon

) that guarantees

|f(x)-7|<\varepsilon

whenever

|x-2|<\delta

. (d) State the final logical sentence that completes the proof that the limit equals

7

. Your sentence should link the existence of the chosen

\delta

to the original definition involving

\varepsilon

. No additional calculations are needed.