Limits and Continuity

Limit Notation and Definition

The concept of a limit is fundamental in calculus and describes the behavior of a function as its input approaches a particular value. The notation for the limit of a function f(x) as x approaches a is:

• Definition: means that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L.

• Example:

Limits of Piecewise Functions

Piecewise functions may have different expressions for different intervals. To find the limit at a point, consider the function's behavior from both sides.

• Example: For defined differently for and , evaluate the limit from the left and right at .

Properties of Limits

Several properties simplify the calculation of limits:

• Limit of a Constant:

• Limit of a Polynomial:

• Limit of a Rational Function: if

• Limit of a Sum/Difference:

• Limit of a Product:

• Limit of a Quotient: if

• Limit of a Power:

• Limit of a Radical:

Limits Involving Infinity

Limits as x approaches infinity often depend on the degree of the numerator and denominator in rational functions.

• Property: (for )

• Property: (for )

• Dominant Term Property: For polynomials, the highest degree term dominates as .

• Cases:

  • Degree of numerator < denominator: limit is 0

  • Degree of numerator = denominator: limit is ratio of leading coefficients

  • Degree of numerator > denominator: limit is infinity or does not exist

Asymptotes

Asymptotes describe the behavior of functions as x approaches certain values.

• Horizontal Asymptote: If , then is a horizontal asymptote.

• Vertical Asymptote: If or , then is a vertical asymptote.

Continuity

A function is continuous at a point if it meets three conditions:

• 1. is defined.

• 2. exists.

• 3.

Polynomials are continuous everywhere. Rational functions are continuous except where the denominator is zero.

Example: For , check continuity at .

Rates of Change

Average Rate of Change

The average rate of change of a function over an interval is:

• Also known as the difference quotient:

Example: For , find the average rate of change on .

Instantaneous Rate of Change

The instantaneous rate of change at is the limit of the average rate of change as the interval shrinks to zero:

• Alternate form:

• This is the slope of the tangent line to the graph at .

The Derivative

Definition of the Derivative

The derivative of a function , denoted , is defined as:

• It represents the slope of the tangent line and the instantaneous rate of change of with respect to .

Example: For , find using the limit definition.

Conditions for Nondifferentiability

A function is not differentiable at a point if:

• 1. The function is not continuous at that point.

• 2. The function has a corner at that point.

• 3. The function has a vertical tangent line or a cusp at that point.