Chapter 2: Limits

2.3 Techniques for Computing Limits

Analytical methods for evaluating limits provide precise results, overcoming the limitations of graphical and numerical estimation. This section introduces foundational theorems and laws for computing limits, especially for polynomial and rational functions.

Limit Laws

Theorem 2: Limit Laws

The following properties simplify the evaluation of 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 near if is even

Limits of Polynomial and Rational Functions

Theorem 4: Limits of Polynomial and Rational Functions

For polynomials and , and constant :

• Polynomial:

• Rational: , provided

Example: To evaluate , apply the limit laws to each term:

This demonstrates that the limit of a polynomial as approaches is simply the value of the polynomial at $a$.

One-Sided Limits

Theorem 7: Limit Laws for One-Sided Limits

Limit laws 1–6 apply to left-sided () and right-sided () limits. For the root law, the function must be non-negative near from the appropriate side if is even.

• Right-sided root law: , provided for

• Left-sided root law: , provided for

Other Techniques for Evaluating Limits

Factoring, Conjugates, and Common Denominators

When direct substitution fails (e.g., results in ), alternative algebraic techniques are used:

• Factoring and canceling common terms

• Multiplying by conjugates to simplify radicals

• Finding common denominators

Example: can be evaluated by multiplying numerator and denominator by the conjugate .

The Squeeze Theorem

Theorem 10: Squeeze Theorem

If for all near (except possibly at $a$), and , then .

This theorem is especially useful when a function is bounded by two others whose limits are known.

Example: Applying the Squeeze Theorem

To verify :

• Since , multiplying by gives .

• Both and .

• By the Squeeze Theorem, .

Trigonometric Limits

Key Trigonometric Limits

Two important trigonometric limits are:

These results allow the evaluation of more complex trigonometric limits using identities and limit laws.

Example: Evaluate and using trigonometric identities such as and .

Summary Table: Limit Laws

Additional info: These notes cover the essential analytical techniques for evaluating limits, including the use of algebraic manipulation, the Squeeze Theorem, and trigonometric identities. Mastery of these concepts is foundational for further study in calculus, including derivatives and integrals.