# Continuity of Polynomial and Rational Functions

1. Let

(a) Replace by

(b) Expand the result so becomes a polynomial in

(c) Replaceby

This way is expressed in the form

(d) Express into the form

(e) Can you find a bound for the function when is near In other words, find a constant which depends on the coefficients and find such that the inequality

holds for all satisfying

(f) Find any such that

for all satisfying

Find any such that

for all satisfying

Find any such that

for all satisfying

(g) For any find such that

for all satisfying

2. Let Given find depending on such that

for all satisfying

3. Let Find such that

for all satisfying

4. Let What is Show that the function is bounded away'' from near . This means: there exists and such that for all satisfying

5. If is bounded away from near show that remains bounded near there exists and such that for all x satisfying

6. Let Given find depending on such that

for all satisfying

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