MTH 252H-Lee Group Project 2 Fall 2004
This project helps build your understanding of the limit concept and what it
means for an limit to exist.
Problem 1 (On Understanding Limits) Let
f(x) = ex1xx2=2
x3
for x6= 0:
(a) Use a calculator to complete the following table:
x101 1011 1012 1012 1011 101
f(x)
Make this a two column table in your report.
(b) Based on the table answer the following questions and give reasons in
support of your answers: Do you think f(x)has a limit as x!0? If yes,
what is it and why? If no, why?
(c) Graph y=f(x)say for xin [1;1] n f0g:Now what do you think about
your answers to (b)? Same? Di¤erent? Why?
Problem 2 (Remember Pythagorus) 4ABC has a right angle at C; AB =c
units, and Dis the midpoint of side BC:
(a) If side BC grows in length tending to in…nity, what can be said of the
limiting value of the ratio AC=AD?
(b) If side BC shrinks in length tending to zero, what can be said of the
limiting value of the ratio AC=AD?
Problem 3 (On Understanding Limits)
(a) Use numerical and/or graphical evidence to evaluate
lim
x!0(1 + x)1=x
correct to two decimal places. Brie‡y explain why you believe you have
achieved two decimal place accuracy.
(b) Can numerical and/or graphical evidence such as you developed in (a)
be used to determine the exact value of limx!0(1 + x)1=x?Explain brie‡y.
(c) Can you be certain from your work in (a) that limx!0(1 + x)1=x exists?
Explain brie‡y.