For , define . Prove that by completing the steps…

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

Problem

For

x\neq -2

, define

f(x)=\dfrac{2x+1}{x+2}

. Prove that

\displaystyle \lim_{x\to 1} f(x)=1

by completing the steps below. (a) Derive an expression for

|f(x)-1|

that isolates a factor of

|x-1|

in the numerator. (b) Explain why the denominator of

f(x)

is bounded away from zero when

|x-1|<1

. Provide a numerical lower bound for

|x+2|

under this assumption. (c) Combine your results from parts (a) and (b) to obtain an inequality of the form

|f(x)-1| \leqslant C|x-1|

, where

C

is a positive constant you must determine. (d) For an arbitrary

\varepsilon>0

, choose an appropriate

\delta

that ensures

|f(x)-1|<\varepsilon

whenever

|x-1|<\delta

and

|x-1|<1

. Conclude the proof with a clear logical statement connecting

\varepsilon

\delta

, and the limit.