Limits, Continuity, and Derivatives

Subject classification: this is a mathematics resource.

Limits and Continuity

Conditions for Continuity

• Conditions for continuity
  • f(c) exists
  • Limit as x approaches c of f(x) exists
  • Limit as x approaches c of f(x) = f(c)
• A limit exists when the right hand and left hand limits are equal to one another

Strategies for Finding Limits

• If the function is continuous, plug in x to find the value of the limit
• If the highest powers of the function are equal, divide their coefficients to find the limit
  • L’Hopital’s Rule: If the limit equals 0/0, differentiate the top and bottom of the fraction until you can plug in the limit

Common Trigonometric Limits

• Limit as x -> 0 of (sinx)/x = 1
• Limit as x -> 0 of (cosx - 1)/x = 0
• Limit as x -> 0 of (sin ax)/x = a
• Limit as x -> 0 of (sin ax)/(sin bx)= a/b

Types of Discontinuities

Jump Discontinuity

• The left-hand and right-hand limits are not equal.
• Both one-sided limits are finite.

Removable Discontinuity

• The limit exists at the point.
• The discontinuity can be removed by rewriting the function (for example, simplifying a rational function by canceling common factors).

Infinite Discontinuity

• At least one of the one-sided limits is infinite (positive or negative).
• This typically occurs at vertical asymptotes.

Limit Definition of a Derivative

The derivative of a function represents the slope of the tangent line at a given point.

Slope = f’(x) = Limit as h -> 0 of ((f(x + h) - f(x)) / (h))