Unit 2: Limits

Question 1

Average Velocity

Answer 1

The average velocity over a time interval

Question 2

Instantaneous Velocity

Answer 2

The velocity at a single instant of time

Question 3

Secant Line

Answer 3

A line joining two points on a curve

Question 4

Limit

Answer 4

As t1 approaches t0, the average velocities typically approach a unique number, which is the instantaneous velocity

Question 5

Tangent Line

Answer 5

The unique line that the secant lines approach

Question 6

Limit of a Function (Preliminary)

Answer 6

Suppose the function f is defined for all x near a except possibly at a. If f(x) is is arbitrarily close to L (as close to L as we like) for all x sufficiently close (but not equal) to a, we write

lim f(x) = L
x - a

and say the limit of f(x) as x approaches a equals L

Question 7

Right-Hand Limit

Answer 7

Suppose f is defined for all x near a with x > a. If f(x) is arbitrarily close to L for all x sufficiently close to a with x > a, we write

lim f(x) = L
x - a+

and say the limit of f(x) as x approaches a from the right equals L.

Question 8

Left-Hand Limit

Answer 8

Supposed f is defined for all x near a with x < a. If f(x) is arbitrarily close to L for all x sufficiently close to a with x < a, we write

lim f(x) = L
x - a-

and say the limit of f(x) as x approaches a from the left equals L

Question 9

Relationship Between One-Sided and Two-Sided Limits Theorem

Answer 9

Assume f is defined for all x near a except possibly at a. Then lim f(x) = L
x - a
if and only if lim f(x) = L and lim f(x) = L
x - a+ x - a-