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

(1)

**Solution:** The notion

(2)

means the value of the function can be as large as we like, 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 simplifying

(9)

4. Is the function

(10)

continuous at

**Solution:** Let us observe that is not defined since

(11)

Therefore, cannot be continuous at Note that in order for a function to be continuous at a point, it must be defined at that point.

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

(12)

It is now easy to see slope is equal to

(13)

and so

(14)

is an equation of the tangent line.

6. Find the limit

(15)

**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

(16)

The conclusion is that

(17)

does not exist.