Explorations 5
Introduction to Limits
1.
Plot on your calculator the graph of this
function.
f(x)
=
x
3

7x
2
+
17x

15
x

3
Use a friendly window with x = 3 as a grid point.
Sketch the results here. Show the behavior of the
function in a neighborhood of x = 3.
2.
Substitute 3 for x in the equation for f(x).
What form does the answer take? What
name is given to an expression of this
form?
3.
The graph of f has a removable
discontinuity at x = 3. The yvalue at this
discontinuity is the limit of f(x) as x
approaches 3. What number does this limit
equal?
4.
Make a table of values of f(x) for each 0.1
unit change in xvalues from 2.5 through
3.5.
5.
Between what two numbers does f(x) stay
when x is kept in the open interval (2.5,
3.5)?
6.
Simplify the fraction for f(x). Solve
numerically to find the two numbers close
to 3 between which x must be kept if f(x) is
to stay between 1.99 and 2.01.
7.
How far from x = 3 (to the left and to the
right) are the two xvalues in Problem 6?
8.
For the statement “If x is within _______
units of 3 (but not equal to 3), then f(x) is
within 0.01 unit of 2,” write the largest
number that can go in the blank.
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 Spring '08
 Grether
 Limits, Limit, Continuous function, Limit of a function, Equals sign, e. f. g.

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