136.169, Assignment No. 2

October 11, 2002

The assignment is due Friday, October 18, 2002 in class. Late assignments receive a mark zero.

1. Find the following limits. Show all of your work.

a) [3]

b) [4]

c) if .[4]

2. Using the formal definition for the limits, prove the following:

a) [5]

b) . [6]

3. a) Find the points of discontinuities for the function ,

and determine if they are removable or not. [6]

b) Find the value for the constant k such that the function

is continuous at x = -1. [4]

c) Give an example of two functions f and g that are not continuous at the point c,

but such that f+g and f.g are both continuous at c. [3]

4. Let K>0 and let f be defined on all of |R. Show that if f satisfies the condition

|f(x) – f(y)| K |x-y| , for all x and y in |R, then f is continuous at every point c

in |R. [7]

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Total [ 42/40]