Section 2.3: Techniques for Computing Limits

Importance of the Section

• Critical for understanding derivatives in Chapter Three.
• Techniques here will be applied in later chapters.
• Two classes dedicated to this section for thorough understanding.

Goal

• Develop algebraic techniques to compute limits by hand.
• Move beyond graphical and numerical methods for exact answers.

Limit Laws

• Linearity Properties:
  • Sum/Difference Law:
    • ( \lim_{{x \to c}} (f(x) \pm g(x)) = \lim_{{x \to c}} f(x) \pm \lim_{{x \to c}} g(x) )
  • Constant Multiple Rule:
    • ( \lim_{{x \to c}} (k \cdot f(x)) = k \cdot \lim_{{x \to c}} f(x) )
• Product Law:
  • ( \lim_{{x \to c}} (f(x) \cdot g(x)) = \lim_{{x \to c}} f(x) \cdot \lim_{{x \to c}} g(x) )
• Quotient Law:
  • ( \lim_{{x \to c}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to c}} f(x)}{\lim_{{x \to c}} g(x)} ) (provided ( \lim_{{x \to c}} g(x) \neq 0 ))

Example Problem

• Given: ( \lim_{{x \to 1}} f(x) = 8 ), ( \lim_{{x \to 1}} g(x) = 3 ), ( \lim_{{x \to 1}} h(x) = 2 )
• Evaluate: ( \lim_{{x \to 1}} \frac{f(x)g(x)}{g(x) - 3h(x)} )
  • Solution: Simplifies to ( -8 ).

Techniques for Evaluating Limits

Direct Substitution

• Definition: If substituting ( c ) into ( f(x) ) results in a real number, the limit is ( f(c) ).
• Works For:
  • Polynomials, exponentials, absolute value, sine, cosine functions.
• Example: ( \lim_{{x \to 0}} \frac{2x-1}{x-2} = \frac{-1}{-2} = \frac{1}{2} )

Factoring and Canceling

• Goal: Simplify expression to eliminate indeterminate forms like ( \frac{0}{0} ).
• Examples:
  • ( \lim_{{x \to 3}} \frac{x^2-9}{x-3} = 6 )
  • ( \lim_{{x \to 2}} \frac{x^2-7x+10}{x-2} = -3 )

Multiplying by Conjugate

• Used When: Square roots involved, result in indeterminate form ( \frac{0}{0} ).
• Property: ((a + b)(a - b) = a^2 - b^2)
• Example:
  • ( \lim_{{h \to 0}} \frac{\sqrt{5h+4} - 2}{h} \rightarrow \frac{5}{4} ).

Conclusion

• This is the first part of Section 2.3, covering fundamental techniques.
• More techniques will be covered in the next class before moving to Section 2.4.