Calculus: Limits flashcards

Question 1

What does multivariable calculus study?

Accepted Answer

Calculus of functions that have two or more variables.

Question 2

What is the slope of the secant line to f through (a, f(a)) and (x, f(x))?

Accepted Answer

m_sec = (f(x) − f(a))/(x − a)

Question 3

What is an object's average velocity over [a, t] if its position is s(t)?

Accepted Answer

v_ave = (s(t) − s(a))/(t − a)

Question 4

If the left and right limits of f at a exist and are equal, what is the limit of f at a?

Accepted Answer

That common value.

Question 5

What is a one-sided limit?

Accepted Answer

A limit approached from just one side: the left or the right.

Question 6

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

Accepted Answer

When the limit of f as x nears a from either side is infinite.

Question 7

What are lim (x→a) x and lim (x→a) c, for a constant c?

Accepted Answer

a and c.

Question 8

What does the sum law for limits state?

Accepted Answer

lim (f(x) + g(x)) = lim f(x) + lim g(x) = L + M

Question 9

What does the constant multiple law for limits state?

Accepted Answer

lim c·f(x) = c · lim f(x) = cL

Question 10

What does the product law for limits state?

Accepted Answer

lim (f(x) · g(x)) = lim f(x) · lim g(x) = L · M

Question 11

What condition does the quotient law for limits require?

Accepted Answer

The limit of the denominator, M, is not 0.

Question 12

What does the squeeze theorem state?

Accepted Answer

If f(x) ≤ g(x) ≤ h(x) for x ≠ a on an open interval containing a, and lim f = L = lim h at a for a real L, then lim (x→a) g(x) = L.

Question 13

What is lim (θ→0) sin θ / θ?

Accepted Answer

1

Question 14

What is lim (θ→0) (1 − cos θ)/θ?

Accepted Answer

0

Question 15

What are the three conditions for f to be continuous at a?

Accepted Answer

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

Question 16

When is f continuous from the right at a?

Accepted Answer

When lim (x→a⁺) f(x) = f(a).

Question 17

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

Accepted Answer

When it is continuous on (a, b), right-continuous at a and left-continuous at b.

Question 18

When does f have a removable discontinuity at a?

Accepted Answer

When f is not continuous at a, yet lim (x→a) f(x) exists.

Question 19

What does the composite function theorem state?

Accepted Answer

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

Question 20

What does the Intermediate Value Theorem state?

Accepted Answer

If f is continuous on [a, b] and z lies between f(a) and f(b), then f(c) = z for some c in [a, b].

Question 21

What does the triangle inequality state?

Accepted Answer

|a + b| ≤ |a| + |b| for all real numbers a and b.

Question 22

What is the epsilon-delta definition of lim (x→a) f(x) = L?

Accepted Answer

For every ε > 0 there is a δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε.

Calculus: Limits flashcards

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