Calculus 1

Continuity at x=a

If f(a) is equal to the limit at x=a.

Limit at x=a

If the limit is the same from both the left and right.

No Limit at x=a

Vertical asymptote or jump discontinuity.

Types of Discontinuity

Removable/hole, infinite, and jump

No Derivative at x=a

If there is a discontinuity, cusp, or vertical tangent.

Limit Definition of Euler’s Number

(1+1/x)^x as x approaches infinity.

Factoring Limits

Cancel common factors, and then substitute.

Substituting Limits

If defined, simply plug in numbers.

Top-Heavy Rational Functions

Oblique asymptote; apply polynomial division.

Equal-Degree Rational Functions

Horizontal asymptote, divide leading coefficients.

Bottom-Heavy Rational Functions

Horizontal asymptote; y=0.

Definition of a Derivative

Limit as h approaches 0 of the difference quotient.

Difference Quotient

Slope formula, but in terms of f(x) and a change of h.

Linearity of Differentiation

Sum Rule and Constant-Multiple Rule: you can differentiate each term individually and you can distribute constants.

Derivative of Cosecant

-csc(x) * cot(x)

Derivative of Cotangent

-csc²(x)

Derivative of Secant

sec(x)tan(x)

Differentiation of Exponential Function a^x

a^x * ln(a)

Derivative of uv (Product Rule)

u’v + uv’

Derivative of u/v (Quotient Rule)

(u’v-uv’)/(v²)

Derivative of f(g(x)) (Chain Rule)

f’(g)*g’

Implicit Differentiation

Differentiate as normal, but append dy/dx to every y term after.

Logarithmic Differentiation

Take the logarithm of both sides and solve using log properties.