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 ax
ax * 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.