Limits

Question 1

The rationalizing technique for evaluating limits

Answer 1

Based on multiplication by a convenient form of 1.

Question 2

Does a limit exist if it equals infinity?

Answer 2

No. Be sure you see that the equal sign in the statement lim f(x) = infinity does not mean that the limit exists! On the contrary, it tells you how the limit fails to exist by denoting the unbounded behavior of f(x) as x approaches c.

Question 3

Where is the following function defined?

y(x) = ln(x-3)

Answer 3

Defined for x > 3 because the natural log is only defined for values greater than 0.

Question 4

Solve for x:

ln (x-3) = 0

Answer 4

e^ln(x-3)= e⁰

x-3 = 1

x=4

* e^lncancels out

Question 5

Conditions for continuity at a point

Answer 5

A function f is continuous at c if these three conditions are met:

1: f(c) is defined
2: The limit of f(x) exists at x=c
3: The limit of f(x) = f(c)

Question 6

Describe continuity on an open interval.

Answer 6

A function is continuous on an open interval (a,b) when the function is continuous at each point in the interval.

Question 7

What is the difference between removable and non-removable discontinuities?

Answer 7

If the limit of the function at the value corresponding to the discontinuity exists, it is removable, if it doesn’t exist it is non-removable. A discontinuity at c is considered removable when a function f can be made continuous by appropriately defining (or redefining) f(c). If the discontinuity can be eliminated by simplifying the expression then the discontinuity is removable.

Question 8

Open and closed interval notation

Answer 8

(1,5) - open
[1,5] - closed

Question 9

What is required for a continuity of a function on a closed interval?

Answer 9

The (one-sided) limit of the function from each direction *(from the + on the left endpoint of the interval, and from the - on the right endpoint of the interval) *must equal the value of the function at that endpoint.

Question 10

Determining whether f(x) approaches ∞ or -∞ as x approaches c from the left and from the right.

Answer 10

In the case of the limit approaching zero, just determine the limit is approaching + zero or -zero. This isn’t accurate, but helpful for estimating. If it is the denominator some number divided by +zero will be +∞.

Question 11

What is x/0?

for any real number x

Answer 11

∞

Question 12

Definition of vertical asymptote

Answer 12

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

Question 13

When using the limit process to find the derivative, what happens to the all of the terms containing h?

Answer 13

They are eliminated because you plug in zero because you are finding the limit as h goes to zero.

Question 14

What exception is required when you simplify a function in order to determine the limit when the limit is not defined in its original form?

Answer 14

You must stipulate that the simplified function is only true for values not equal to the value that resulted in zero in the denominator/numerator of the original function.

Question 15

Continuity

Answer 15

Remember to be continuous at any point the function must equal the limit at that point from whatever direction is relevant to the question.

Question 16

Intermediate Value Theorem

Answer 16

For the theorem to apply, you have to make sure the function is continuous over the given interval, and you also have to make sure that the function at the interval endpoints are not equal. If both are not true, then the theorem doesn’t apply to the situation. Then set the requesteed value f(c) equal to the function, and if it is within the given interval that is/are the value of f(c) which is guaranteed by the theorem. This is stupid and easy if you remember what it mean (because if a line is continuous between two points, there is going to be a guaranteed value for the function in between the points). If it says f(c) = 11, you plug 11 into the equation if all the conditions are met. Book page 97

Question 17

When evaluating limits with absolute values to eliminate confusion:

Answer 17

abs(x-5) = 5-x when evaluating the limit as x->c from the left and x-5 from the right. The rearranged expression from the left will always be equivalent, and less confusing. This is the same as saying f(x) = 5-x for x < 5