exists: by the basic properties of limits we have 
1. (15 points) Use the definition of continuity to prove that the function f(x) = 3 x + 1 is continuous at every point on the real line.
Solution Let x = a be an arbitrary point on the real line. We check the three conditions for continuity:
1) f(a) is defined: f(a) = 3 a + 1
2) exists: by the basic properties of limits we have
3) 
: follows from parts 1 and 2.
2. (25 points) Sketch the graph of the function
Be sure to include information about horizontal and vertical asymptotes.
Solution Factoring the numerator and denominator we have
Everywhere except for x = -1 (where the function is undefined), this is the same function as
This function has a vertical asymptote at x = 1. For x just to the left of 1, 
small negative), which means that the function dives off toward 
as we approach 1 from the left. Similarly, for x just to the right of 1, 
small positive), so y goes to 
as we approach from the right.
In the limit of large x, the function behaves more and more like y = x/x = 1, so the function has a horizontal asymptote at y = 1.
The numerator vanishes at x = -2. That is the only point where the graph crosses the x-axis.
Here is the graph:
 |
 |
(Note also that the graph should contain a hole at x = -1, y = -1/2.)
3. (10 points) Solve the equation
for x.
Solution Dividing both sides by ex gives
Taking a natural logarithm on both sides gives
x - 4 = ln 2
x = 4 + ln 2
4. (20 points) Find the equation of the tangent line to the curve y = x2 + 2 at the point (1, 3).
Solution To compute the slope of the tangent line at the point x = 1 we compute
The equation of the tangent line is y = m x + b = 2 x + b. To solve for b we substitute the point (1,3).
3 = 2 (1) + b
b = 1
The equation of the tangent line is
y = 2 x + 1
5. (15 points) Compute the limit
Solution
6. (15 points) State the intermediate value theorem.
Solution Let f(x) be continuous throughout the closed interval [a,b]. Let m be any number between f(a) and f(b). Then there is at least one c in [a,b] such that f(c) = m.