Question

1. The point lies on the curve . (a) , use your calculator to find the

slope of the secant line (correct to six decimal places)

for the following values of x:

(i) 1.5 (ii) 1.9

(iii) 1.99 (iv) 1.999

(v) 2.5 (vi) 2.1

(vii) 2.01 (viii) 2.001

(b) Using the results of part (a), guess the value of the slope of

the tangent line to the curve at P2, ln 2.

PQ

If Q is the point x, ln x

P2, ln 2 y ln x

(c) Using the slope from part (b), find an equation of the

tangent line to the curve at P2, ln 2.

(d) Sketch the curve, two of the secant lines, and the tangent

line.

2. Use the given graph of f to state the value of each quantity,

if it exists. If it does not exist, explain why.

(a) lim (b) f x (c)

(d) (e)

y

0 2 4 x

4

2

f 5 xl5

lim fx

lim

xl1

lim fx xl1 xl1 f x

3. Sketch the graph of the following function and use it to determine the values

of a for which exists:

f x

2 x

x

x 1

2

if x 1

if 1
x 1

if x 1

limxla fx

4. Determine the infinite limit.

(a) (b) lim

xl5

6

x 5

lim

xl5

6

x 5

(c)

x l 0

lim x 1

x 2

x 2 (d) lim

x l2

x 1

x 2

x 2

5. (a) Find the vertical asymptotes of the function

(b) Confirm your answer to part (a) by graphing the function.