Limits (Chapter 2)

Question 1

What is the process for evaluating limits analytically?

Answer 1

1. Attempt direct substitution
2. If direct substitution doesn’t work, use algebra and factor as much as possible, removing like terms from numerator and denominator.

Question 2

What is the epsilon delta definition of a limit?

Answer 2

Lim(x->c) f(x) = L means that for each epsilon > 0 there exists a delta > 0 such that if 0 < |x-c| < delta then |f(x)-L| < epsilon

Question 3

How can you find the slope of a secant line between two points?

Answer 3

f(c + delta x) - f(c) / delta x

Question 4

Under what conditions does a limit fail to exist?

Answer 4

1. Limit on the left side is different from limit on the right side
2. Limit increases or decreases without bound as x approaches c
3. Limit oscillates between two fixed values as x approaches c

Question 5

What is the expression 0/0 considered?

Answer 5

Indeterminate form

Question 6

What is the squeeze theorem?

Answer 6

If h(x) c) h(x) = L = lim(x->c) g(x) then lim(x->c) f(x) exists and is equal to L.

Question 7

What are the three special limits?

Answer 7

Lim(x->0) sin x / x = 1
Lim(x->0) 1-cosx / x = 0
Lim(x->0) (1 + x) ^ 1 / x = e

Question 8

What are the requirements for a function to be continuous at a given point c?

Answer 8

1. f(c) is defined
2. Lim (x->c) f(x) exists
3. Lim (x->c) f(x) = f(c)

Question 9

How can a removable discontinuity be removed?

Answer 9

By appropriately redefining or defining f(c) by factoring or other methods

Question 10

What is the greatest integer function?

Answer 10

The greatest integer n such that n <= x

Question 11

What is the intermediate value theorem?

Answer 11

If f is continuous on [a,b] and f(a) != f(b), and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k.

Question 12

What is an infinite limit?

Answer 12

A limit in which f(x) increases or decreases without bound as x approaches c.

Question 13

What is a vertical asymptote?

Answer 13

If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f.

Question 14

What are the properties of infinite limits?

Answer 14

1. Infinity +/- Limit = Infinity
2. Infinity * Limit = Infinity, provided Limit > 0
3. Infinity * Limit = (neg)Infinity, provided Limit < 0
4. Limit / Infinity = 0

Question 15

What are the properties of negative infinite limits?

Answer 15

1. Negative Infinity +/- Limit = Negative Infinity
2. Negative Infinity * Limit = Negative Infinity, provided Limit > 0
3. Negative Infinity * Limit = (pos)Infinity, provided Limit > 0
4. Limit / Negative Infinity = 0