Techniques for Computing Limits

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding how to compute limits is essential for studying derivatives, integrals, and continuity.

Basic Limit Laws

Direct Substitution and Limit Laws

For many linear, polynomial, and fully simplified rational functions, limits can be evaluated using direct substitution. The following limit laws simplify the evaluation of many limits. Assume and exist, is a real number, and is an integer:

• Sum Law:

• Difference Law:

• Constant Multiple Law:

• Product Law:

• Quotient Law: , provided

• Power Law:

• Root Law: , provided for even

Examples Using Limit Laws

Applying Limit Laws

• Quotient Law Example:

• Constant Multiple Law Example:

• Product and Root Law Example:

Limits of Constant and Polynomial Functions

Constant Functions

• The limit of a constant function is always equal to the value of the constant.

• Example:

Polynomial Functions

• For polynomials, direct substitution can be used to evaluate the limit.

• Example:

Limits Involving Rational Functions

Direct Substitution

• If direct substitution does not produce division by zero, the limit can be evaluated directly.

• Example:

Factor and Cancel Technique

• If direct substitution results in division by zero, factor and simplify the expression before evaluating the limit.

• Example:

Special Techniques for Limits

Combining Terms with Common Denominators

• When expressions involve sums or differences of rational functions, find a common denominator before evaluating the limit.

• Example:

Using Conjugates

• For limits involving square roots, multiply by the conjugate to simplify the expression.

• Example:

Trigonometric Identities

• Use trigonometric identities to simplify limits involving trigonometric functions.

• Example:

Limits That Do Not Exist

Piecewise Functions

• For piecewise functions, evaluate the left-hand and right-hand limits separately. If they are not equal, the limit does not exist.

• Example: For :

  • does not exist (dne) because the left and right limits are not equal.

Summary Table: Limit Laws

Additional info: These notes cover the main techniques for evaluating limits, including direct substitution, factoring, rationalizing, and using trigonometric identities. Mastery of these techniques is essential for further study in calculus, especially for understanding continuity, derivatives, and integrals.