Consider the function

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

Problem

f(x)=x^{2}+1

. Show that

\displaystyle \lim_{x\to 3} f(x)=10

by establishing an explicit relationship between

\varepsilon

and

\delta

. (a) Factor and simplify

|f(x)-10|

to obtain an expression that involves

|x-3|

and

|x+3|

. (b) Restrict attention to values of

x

satisfying

|x-3|<1

. Under this restriction, determine an upper bound for

|x+3|

and use it to produce an inequality of the form

|f(x)-10| \leqslant K|x-3|

, where

K

is a positive constant you must specify. (c) For a given

\varepsilon>0

, choose a positive

\delta

that depends on

\varepsilon

and the constant

K

so that

|f(x)-10|<\varepsilon

whenever

|x-3|<\delta

and

|x-3|<1

. State the explicit formula for

\delta

. (d) Justify why the additional condition

|x-3|<1

does not limit the generality of the proof, and conclude that the desired limit equals

10

. Provide a concise logical explanation rather than further algebra.