Continuity of Polynomial and Rational Functions

1. Let MATH

(a) Replace $x$ by $y+2.$

(b) Expand the result so $h$ becomes a polynomial in $y.$

(c) Replace$\;y\;$by $x-2.$

This way $h(x)$ is expressed in the form
MATH

(d) Express $h(x)-h(2)$ into the form
MATH

(e) Can you find a bound for the function $g(x)$ when $x$ is near $2?$ In other words, find a constant $M>0\;$which depends on the coefficients $a,b,c,d,e,f$ and find $\delta >0\;$such that the inequality
MATH
holds for all $x$ satisfying
MATH

(f) Find any $\delta _1>0$ such that
MATH
for all $x$ satisfying
MATH
Find any $\delta _2>0$ such that
MATH
for all $x$ satisfying
MATH
Find any $\delta _3>0$ such that
MATH
for all $x$ satisfying
MATH

(g) For any $\epsilon >0,$ find $\delta >0$ such that
MATH
for all $x$ satisfying
MATH

2. Let MATH Given $\epsilon >0,$ find $\delta >0$ depending on $\epsilon $ such that
MATH
for all $x$ satisfying
MATH

3. Let MATH Find $\delta >0$ such that
MATH
for all $x$ satisfying MATH

4. Let $f(x)=x^2-4x+1.$ What is $f(2)?\;$Show that the function $f(x)$ is ``bounded away'' from $0$ near $2$. This means: there exists $m>0$ and $\delta >0\;$such that MATH for all $x$ satisfying MATH

5. If $f(x)$ is bounded away from $0$ near $x=a,$ show that $\frac 1{f(x)}$ remains bounded near $x=a:$there exists $M>0$ and $\delta >0\;$such that MATH for all x satisfying MATH

6. Let MATHGiven $\epsilon >0,$find $\delta >0$ depending on $\epsilon $ such that
MATH
for all $x$ satisfying MATH

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