1. Suppose is a given function and and are real numbers. State what we mean by

(1)

**Solution:** The notion

(2)

means the value of the function can be as close as we like to the number , whenever is close enough to the number

2. Does the limit

(3)

exist?

**Solution:** We know when and when We write

(4)

Next,

(5)

Now since

(6)

we can conclude

(7)

does not exist.

3. Compute the limit

(8)

**Solution:** We start by factoring and then simplifying

(9)

4. Is the function

(10)

continuous at

**Solution:** We note that

(11)

Also

(12)

It is easy to see that

(13)

and therefore the function is continuous at

5. Find an equation of the tangent line to the graph of at

**Solution:** In order to write the point-slope formula of any line, we need a point on the line and the slope of the line. The point is given . To find the slope, we compute the derivative of since evaluating the derivative at the given point provides the slope of the tangent line at that point. Write

(14)

Therefore the slope is found to be

(15)

and an equation of the tangent line is

(16)

6. Find the limit

(17)

**Solution:** We immediately notice that denominator approaches zero as approches Since the sign of denominator will be different as approaches from right and left, we obtain

(18)

The conclusion is that

(19)

does not exist.