Limits

Question 1

Tangent

Answer 1

The tangent to a curve is a line that touches the curve and has the same direction as the curve at the point of contact

Question 2

Secant

Answer 2

A secant line to a curve is a line that intersects (or ‘cuts’) the curve more than once.

Question 3

the limit of the slopes of the secant lines as we move a second point closer to a set point P

Answer 3

The slope of the tangent line at P

Question 4

How do you say lim(x→a) f(x) = L ?

Answer 4

“the limit of f(x), as x approaches a, equals L”

Question 5

What point is not considered when finding the limit of f(x) at x = a ?

Answer 5

in finding the limit of f(x) at x = a we never consider x = a

Question 6

When finding the limit of f(x) at x = a does the function need to be defined at x = a ?

Answer 6

f(x) need not even be defined when x = a.

Question 7

A Limit

Answer 7

Suppose f(x) is defined when x is near the number a. Then we write
lim(x→a) f(x) = L

Question 8

How can the limits of a function be estimated?

Answer 8

The limit of a function can be estimating by making the values of f(x) arbitrarily close to L by restricting x to be sufficiently close to a (on either side of a) but not equal to a.

Question 9

The left-hand limit

Answer 9

lim(x →a−) f(x)=L

Question 10

The right-hand limit

Answer 10

lim(x →a+) f(x) = L

Question 11

How do you say lim(x →a+) f(x) = L ?

Answer 11

the right-hand limit of f(x), as x approaches a is equal to L
OR
the limit of f(x) as x approaches a from the right is equal to L

Question 12

How do you say lim(x →a−) f(x)=L ?

Answer 12

the left-hand limit of f(x), as x approaches a is equal to L
OR
the limit of f(x) as x approaches a from the left is equal to L

Question 13

What are the conditions for the left and right hand limits required for the limit to exist?

Answer 13

lim(x →a−) f(x) = L = lim(x →a+) f(x)

Question 14

What does lim(x →a) f(x) = ∞ mean?

Answer 14

the values of f(x) can be made arbitrarily large by taking x sufficiently close to a, but not equal to a.

Question 15

Infinite limits

Answer 15

lim(x →a) f(x) = ∞
AND
lim(x →a) f(x) = -∞

Question 16

The vertical line x = a is called a vertical asymptote of the curve y = f(x) if at least one of these statements is true

Answer 16

lim(x →a) f(x) = ∞
lim(x →a) f(x) = -∞
lim(x →a-) f(x) = ∞
lim(x →a-) f(x) = -∞
lim(x →a+) f(x) = ∞
lim(x →a+) f(x) = -∞

Question 17

Limit sum law

Answer 17

lim(x→a) [f(x)+g(x)] = lim(x→a) f(x) + lim(x→a) g(x)

Question 18

Condition for the limit laws

Answer 18

Limits lim(x→a) f(x)  and lim(x→a) g(x) exist
c is a constant
n is a positive integer

Question 19

Limit difference law

Answer 19

lim(x→a) [f(x) - g(x)] = lim(x→a) f(x) - lim(x→a) g(x)

Question 20

Limit constant multiple law

Answer 20

lim(x→a) [cf(x)] = c lim(x→a) f(x)

Question 21

Limit Product Law

Answer 21

lim(x→a) [f(x)g(x)] = lim(x→a) f(x) · lim(x→a) g(x)

Question 22

Limit quotient law

Answer 22

lim(x→a) [f(x)/g(x)] = [lim(x→a) f(x)] / [lim(x→a) g(x)]

Question 23

Condition for the limit quotient law

Answer 23

lim(x→a) g(x) does not equal 0

Question 24

Limit power law

Answer 24

lim(x→a) [f(x)]^n = [lim(x→a) f(x)]^n

Question 25

Limit root law

Answer 25

lim(x→a) nth root[f(x)]= nth root[lim(x→a) f(x)]

Question 26

Limits Direct substitution property

Answer 26

If f is a polynomial and a is the domain of f, then

lim(x →a) f(x) = f(a)

Question 27

Squeeze theorem

Answer 27

If f(x) ≤ g(x) ≤ h(x) when x is near a (except maybe at a) and
lim(x →a) f(x) = lim(x →a) h(x) = L
then
lim(x →a) g(x) = L

Question 28

What does the squeeze theorem mean?

Answer 28

if g(x) is squeezed between f(x) and h(x) near a ,and if f and h have the same limit L at a, then g is forced to also have the same limit L at a.

Question 29

What makes a function continuous at a number a?

Answer 29

lim(x →a) f(x) = f(a)

Question 30

What makes a function discontinuous at a?

Answer 30

It is not continuous at a