Limits and Calculus Flashcards

Flashcards based on limits and calculus rules.

Limit of a Constant

For any real numbers a and c, the limit as x approaches a of c equals c.

Limit of a Power Function

For any real numbers a and n, the limit as x approaches a of x^n equals a^n.

lim (x->0) sin(x)/x

The limit as x approaches 0 of sin(x)/x equals 1.

lim (x->0) (1-cos(x))/x

The limit as x approaches 0 of (1 - cos(x))/x equals 0.

lim (x->0) arcsin(x)/x

The limit as x approaches 0 of arcsin(x)/x equals 1.

lim (x->0) (e^x-1)/x

The limit as x approaches 0 of (e^x - 1)/x equals 1.

lim (x->0) (a^x-1)/x, a>0

For a > 0, the limit as x approaches 0 of (a^x - 1)/x equals ln(a).

lim (x->inf) x^a / b^x, b>1

For b > 1 and a any real number, the limit as x approaches infinity of x^a / b^x equals 0.

lim (x->0+) x^x

The limit as x approaches 0 from the right of x^x equals 1.

lim (x->0+) x ln x

The limit as x approaches 0 from the right of x * ln(x) equals 0.

lim (x->inf) ln(x)/x

The limit as x approaches infinity of ln(x) / x equals 0.

lim (x->inf) x^(1/x)

The limit as x approaches infinity of x^(1/x) equals 1.

lim (x->inf) (1 + 1/x)^x

The limit as x approaches infinity of (1 + 1/x)^x equals e.

lim (x->0) (1 + x)^(1/x)

The limit as x approaches 0 of (1 + x)^(1/x) equals e.

lim (x->0) (1 + sin x)^(1/x)

The limit as x approaches 0 of (1 + sin(x))^(1/x) equals e.

lim (x->inf) (x - sqrt(x^2 - a^2))

The limit as x approaches infinity of (x - sqrt(x^2 - a^2)) equals 0.

lim (x->a) (x^n - a^n)/(x-a)

For a > 0 and n a positive integer, the limit as x approaches a of (x^n - a^n)/(x-a) equals n*a^(n-1).

lim (x->0) (tan x - sin x) / x^3

The limit as x approaches 0 of (tan(x) - sin(x)) / x^3 equals 1/2.

lim (x->inf) (log x)^p / x^q, q>0

For q > 0, the limit as x approaches infinity of (log x)^p / x^q equals 0.

tan(x + pi/2) Identity

tan(x + pi/2) = -cot(x)

Limit of (c + z/n)^n as n approaches infinity

For a real or complex constant c and a variable z, limit as n approaches infinity of (c + z/n)^n = e^(cz)

Limit of n(x^(1/n) - 1) as n approaches infinity

For x real (or complex), limit as n approaches infinity of n(x^(1/n) - 1) = log x