MATH 140: Exam 1 Monday, September 22, 2008
Show all work and justify your answers. Your solutions should read nicely and be legible.
They should not be composed of regurgitated fragments of your mind scattered about the
page.
1. Determine which of the following limits exist as real numbers, do not exist, or exist as
โor โโ. If the limit exists, evaluate it. Give reasons and show intermediate steps. [10 pts each]
(a) lim
xโ4โ
x2+xโ20
x2โ8x+ 16
(b) lim
xโฯ/4โptan(x)โ1
(c) lim
xโ1(ln(x))2sin ๎1
xโ1๎
(d) lim
xโ4+|8โ2x|
xโ4
(e) lim
xโ1
e2xโ2โ1
xโ1
2. Let f(x) = โx+ 1
(a) By evaluating an appropriate limit, find f0(3) [10 pts]
(b) Write the equation of the tangent line to f(x) at x= 3. [5 pts]
3. Let f(x) = 1 โx3. By evaluating an appropriate limit, find f0(x). [10 pts]
4. Suppose you want to apply the bisection to approximate d, where d=3
โ10.
(a) Write down a function, f(x) that has a das an x-intercept. [5 pts]
(b) For f(x) from part (a), determine an interval on which f(x) has an x-intercept.
State the Theorem that you used to make this determination AND why it was
applicable. (i.e. What conditions on f(x) enabled you to apply this theorem?) [5 pts]
(c) Using your results from parts (a) and (b), apply the bisection method to approx-
imate 3
โ10 with an error <1/4. [10 pts]
5. Use the Definition of a limit to complete the following:
โ lim
xโ3ฯ/4tan(x) = โ1โ means โfor any ๎ > 0. . .โ [5 pts]