Unit 1: Limits and Continuity

Question 1

How do find a limit graphically?

Answer 1

Trace your finger along the graph from left and right and see what the y-value gets close to. Where does the road go at a certain x-value? Ignore what is actually happening there (the bridge).

Question 2

What is the difference between f(a) and the limit as x approaches a of f(x)?

Answer 2

f(a) is what the function actually equals at x=a (the bridge). The limit is what the function is getting close to around x = a.

Question 3

How do you find a limit using a table?

Answer 3

Plug in numbers close on either side (like .999 and 1.001) and see if the y-values are getting close to one specific number.

Question 4

What does it mean for a function to be continuous?

Answer 4

You could draw it without picking up your pencil. The limit always equals the function value.

Question 5

If the limit as x approaches a exists but f(a) DNE, what is going on?

Answer 5

There is a removable point discontinuity (hole) at x = a.

Question 6

If the limit as x approaches a from the left doesn’t match the limit from the right, what is going on?

Answer 6

There is a jump in the function at x = a.

Question 7

If the limit as x approaches a from at least one side approaches infinity what is going on?

Answer 7

There is a vertical asymptote at x = a.

Question 8

How do you tell if a function (usually a piecewise) is continuous at x = a?

Answer 8

3 parts: show that the limit from the left and right match. Show that f(a) also matches. State that f(x) is continuous at a because all three are equal.

Question 9

When you see “find a limit” or a limit = ? the answer should be…

Answer 9

A number or possibly an infinity or DNE

Question 10

When you see “must there be” or “does there exist” you think…

Answer 10

Use IVT or MVT. You’ll need to show some appropriate justifications and then restate the question in your answer

Question 11

When you see “average rate of change” you think…

Answer 11

ALGEBRA. Find a slope over an interval. On a graph, it’s the slope of a secant line.

Question 12

When you see “instantaneous rate of change” you think

Answer 12

Derivative at that point or the slope (steepness) of the tangent line at that point

Question 13

When you see “find the vertical asymptotes of f(x)” you think

Answer 13

Set denominator equal to zero and solve

Question 14

When you see “find the horizontal asymptotes of f(x)” you think

Answer 14

If f(x) is rational, compare top and bottom growth rates.