AP Calculus: Limits flashcards

Question 1

What name is given to a limit taken only from the left or only from the right of a point?

Accepted Answer

A one-sided limit.

Question 2

How is a one-sided limit from the left written for f at a, with the limit value L?

Accepted Answer

lim x->a- f(x) = L

Question 3

How is a one-sided limit from the right written for f at a, with the limit value L?

Accepted Answer

lim x->a+ f(x) = L

Question 4

If the limits from the left and from the right at a point both exist and are equal, what follows?

Accepted Answer

The limit of the function at that point exists and equals that common value.

Question 5

What does it mean for a function to have an infinite limit at a point a?

Accepted Answer

The function increases or decreases without bound as x approaches a.

Question 6

When does a function have a vertical asymptote at x = a?

Accepted Answer

When a one-sided limit at a (from the right, from the left, or both) is infinite.

Question 7

What is the two-sided infinite limit lim x->a f(x) = +infinity in terms of one-sided limits?

Accepted Answer

Both lim x->a- f(x) = +infinity and lim x->a+ f(x) = +infinity.

Question 8

State the constant multiple law for limits.

Accepted Answer

lim x->a c f(x) = c * lim x->a f(x) = cL, where c is a constant.

Question 9

State the quotient law for limits, with lim f(x) = L and lim g(x) = M as x->a, and give its condition.

Accepted Answer

lim x->a f(x)/g(x) = L/M, provided M is not 0.

Question 10

State the power law for limits, with lim f(x) = L as x->a.

Accepted Answer

lim x->a (f(x))^n = L^n, for every positive integer n.

Question 11

What does the squeeze theorem conclude about g, given f(x) <= g(x) <= h(x) near a, and lim f = L = lim h as x->a?

Accepted Answer

lim x->a g(x) = L. The inequality must hold near a (for x not equal to a, on an open interval around a), and L must be a real number.

Question 12

What three conditions make f continuous at a point a?

Accepted Answer

f(a) is defined, lim x->a f(x) exists, and lim x->a f(x) = f(a).

Question 13

When is a function continuous from the left at b?

Accepted Answer

When lim x->b- f(x) = f(b).

Question 14

When is a function continuous from the right at a?

Accepted Answer

When lim x->a+ f(x) = f(a).

Question 15

When is f continuous over a closed interval [a, b]?

Accepted Answer

When it is continuous at every point of (a, b), continuous from the right at a, and continuous from the left at b.

Question 16

What is a removable discontinuity at a point a?

Accepted Answer

A point where f is discontinuous but lim x->a f(x) exists.

Question 17

What is a jump discontinuity at a point a?

Accepted Answer

A point where both one-sided limits exist but are not equal.

Question 18

What does the Intermediate Value Theorem say, and under what condition?

Accepted Answer

If f is continuous on the closed bounded interval [a, b] and z lies between f(a) and f(b), some c in [a, b] gives f(c) = z.

Question 19

Complete the epsilon-delta definition: lim x->a f(x) = L if for every epsilon > 0 there is a delta > 0 such that ...

Accepted Answer

...if 0 < |x - a| < delta, then |f(x) - L| < epsilon.

Question 20

State the composite function theorem.

Accepted Answer

If f is continuous at L and lim x->a g(x) = L, then lim x->a f(g(x)) = f(lim x->a g(x)) = f(L).

Question 21

What are the values of lim sin(theta)/theta and lim (1 - cos(theta))/theta as theta->0?

Accepted Answer

The first is 1 and the second is 0.

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