2.3 Limits: Laws, Substitution, and Squeeze

Limit Laws Overview

• Sum Law: $\lim<em>{x\to a}[f(x)+g(x)] = \lim</em>{x\to a}f(x) + \lim_{x\to a}g(x)$

• Difference Law: $\lim<em>{x\to a}[f(x)-g(x)] = \lim</em>{x\to a}f(x) - \lim_{x\to a}g(x)$

• Constant Multiple Law: $\lim<em>{x\to a}[c\,f(x)] = c\,\lim</em>{x\to a}f(x)$

• Product Law: $\lim<em>{x\to a}[f(x)g(x)] = \left(\lim</em>{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right)$

• Quotient Law: $\lim<em>{x\to a}\frac{f(x)}{g(x)} = \frac{\lim</em>{x\to a}f(x)}{\lim<em>{x\to a}g(x)}\quad(\text{provided } \lim</em>{x\to a}g(x) \neq 0)$

• Power Law: \lim{x\to a}[f(x)]^{n} = \left(\lim{x\to a}f(x)\right)^{n},\quad n\in\mathbb{Z}_{>0}

• Root Law: \lim{x\to a}[f(x)]^{1/n} = \left(\lim{x\to a}f(x)\right)^{1/n},\quad n\in\mathbb{Z}_{>0}

• Special limits:\n • $\lim<em>{x\to a} c = c$\n • \lim{x\to a} x^{n} = a^{n},\quad n\in\mathbb{Z}_{>0}

• If $\lim<em>{x\to a} f(x)=L$, then $\lim</em>{x\to a} [f(x)]^{n} = L^{n}$ and $\lim_{x\to a}[f(x)]^{1/n} = L^{1/n}$ (for appropriate domain, especially when even roots are involved)

Direct Substitution and Continuity

• Direct Substitution Property: If f is a polynomial or a rational function and a is in the domain of f, then$\lim_{x\to a} f(x) = f(a)$
• Continuity: such functions are continuous at a
• If substitution yields an indeterminate form like 0/0, use algebraic simplification (e.g., factoring, canceling common factors) to proceed

One-Sided Limits

• Theorem: $\lim<em>{x\to a} f(x) = L$ exists iff both one-sided limits exist and are equal:$\lim</em>{x\to a^-} f(x)=\lim_{x\to a^+} f(x)=L$
• One-sided limit laws apply in the same way as two-sided limits

The Squeeze Theorem

• If $f(x) \le g(x) \le h(x)$ near a (except possibly at a) and $\lim<em>{x\to a} f(x)=\lim</em>{x\to a} h(x)=L$, then $\lim_{x\to a} g(x)=L$
• Also called the Sandwich Theorem/Squeeze Theorem
• Example form: if $-1 \le \sin(1/x) \le 1$, then for any nearby a where x→0,$-x^{2} \le x^{2}\sin(1/x) \le x^{2}$ and since $\lim<em>{x\to 0}(-x^{2})=\lim</em>{x\to 0}(x^{2})=0$, we get $\lim_{x\to 0} x^{2}\sin(1/x)=0$

Evaluating Limits by Direct Substitution

• If f is a polynomial or rational and a is in the domain, then direct substitution yields the limit
• If not directly substitutable, use algebraic manipulation (e.g., factorization, cancellation) to remove removable discontinuities

Quick Reference Rules (Summary)

• If a limit of f and g exists, the limit of sums, differences, products, and quotients (with nonzero denominator) follows the corresponding algebraic rule
• If the inner function tends to L, powers and roots carry through as per Power/Root Laws
• For limits of composite expressions, reduce to a form where the limit laws can be applied

Example Highlights (Memorandum)

• When evaluating limits graphically or via algebra:
  • Use Sum, Difference, Product, Quotient Laws to combine known limits
  • Use Direct Substitution for polynomials and rational functions once feasible
  • Use the Squeeze Theorem for products with bounded oscillations, e.g., limits involving sin(1/x)