5.3 Limit Theorems for Functions

Limit theorems for functions are crucial tools in calculus. They help us understand how functions behave as we approach specific points or infinity. These rules simplify complex limit calculations by breaking them down into smaller, more manageable parts.

By mastering these theorems, you'll be able to tackle a wide range of limit problems. From basic arithmetic operations to more advanced techniques like L'Hôpital's rule, these tools form the foundation for understanding continuity, derivatives, and integrals in calculus.

Limit Laws for Function Operations

Sums and Differences

• The limit of a sum of two functions equals the sum of their limits, provided both limits exist
  • Mathematically: $\lim[f(x) + g(x)] = \lim f(x) + \lim g(x)$
  • Example: If $\lim f(x) = 3$ and $\lim g(x) = 5$, then $\lim[f(x) + g(x)] = 3 + 5 = 8$
• The limit of a difference of two functions equals the difference of their limits, provided both limits exist
  • Mathematically: $\lim[f(x) - g(x)] = \lim f(x) - \lim g(x)$
  • Example: If $\lim f(x) = 7$ and $\lim g(x) = 2$, then $\lim[f(x) - g(x)] = 7 - 2 = 5$

Products, Quotients, and Powers

• The limit of a product of two functions equals the product of their limits, provided both limits exist
  • Mathematically: $\lim[f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x)$
  • Example: If $\lim f(x) = 4$ and $\lim g(x) = 6$, then $\lim[f(x) \cdot g(x)] = 4 \cdot 6 = 24$
• The limit of a quotient of two functions equals the quotient of their limits, provided the limit of the denominator is non-zero
  • Mathematically: $\lim[f(x) / g(x)] = \lim f(x) / \lim g(x)$, where $\lim g(x) \neq 0$
  • Example: If $\lim f(x) = 10$ and $\lim g(x) = 2$, then $\lim[f(x) / g(x)] = 10 / 2 = 5$
• The limit of a constant multiple of a function equals the constant multiple of the limit of the function
  • Mathematically: $\lim[c \cdot f(x)] = c \cdot \lim f(x)$, where $c$ is a constant
  • Example: If $\lim f(x) = 3$ and $c = 4$, then $\lim[c \cdot f(x)] = 4 \cdot 3 = 12$
• The limit of a function raised to a power equals the limit of the function raised to that power, provided the limit exists
  • Mathematically: $\lim[f(x)^n] = [\lim f(x)]^n$, where $n$ is a real number
  • Example: If $\lim f(x) = 2$ and $n = 3$, then $\lim[f(x)^n] = 2^3 = 8$

Evaluating Limits of Functions

Polynomial, Rational, and Exponential Functions

• To find the limit of a polynomial function, evaluate the function at the point of interest by substituting the limiting value for the variable
  • Example: If $f(x) = 3x^2 + 2x - 1$, then $\lim_{x \to 2} f(x) = 3(2)^2 + 2(2) - 1 = 15$
• To find the limit of a rational function, factor and cancel common factors in the numerator and denominator, then evaluate the simplified function at the point of interest
  • Example: If $f(x) = \frac{x^2 - 4}{x - 2}$, then $\lim_{x \to 2} f(x) = \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 4$
• To find the limit of an exponential function, use the properties of exponents and evaluate the function at the point of interest
  • Example: If $f(x) = 3e^{2x}$, then $\lim_{x \to 1} f(x) = 3e^{2(1)} = 3e^2 \approx 22.17$

Logarithmic Functions and Indeterminate Forms

• To find the limit of a logarithmic function, use the properties of logarithms and evaluate the function at the point of interest
  • Example: If $f(x) = \ln(x^2 + 1)$, then $\lim_{x \to 0} f(x) = \ln(0^2 + 1) = \ln(1) = 0$
• When evaluating limits, be aware of potential indeterminate forms, such as $0/0$, $\infty/\infty$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, and $\infty^0$, which may require further manipulation using L'Hôpital's rule or other techniques
  • Example: If $f(x) = \frac{x^2 - 1}{x - 1}$, then $\lim_{x \to 1} f(x)$ is an indeterminate form of type $0/0$ and can be evaluated using L'Hôpital's rule or factoring

The Squeeze Theorem for Limits

Applying the Squeeze Theorem

• The Squeeze Theorem states that if $f(x) \leq g(x) \leq h(x)$ for all $x$ near $a$, except possibly at $a$, and if $\lim f(x) = \lim h(x) = L$ as $x \to a$, then $\lim g(x) = L$ as $x \to a$
  • Example: If $0 \leq \sin x \leq x$ for all $x > 0$ and $\lim_{x \to 0} 0 = \lim_{x \to 0} x = 0$, then by the Squeeze Theorem, $\lim_{x \to 0} \sin x = 0$
• To apply the Squeeze Theorem, find two functions, $f(x)$ and $h(x)$, that "squeeze" the given function $g(x)$ from below and above, respectively, near the point of interest
  • Example: To find $\lim_{x \to 0} x^2 \cos(\frac{1}{x})$, we can use the inequality $-1 \leq \cos(\frac{1}{x}) \leq 1$ and squeeze $x^2 \cos(\frac{1}{x})$ between $-x^2$ and $x^2$
• Show that the limits of the squeezing functions, $f(x)$ and $h(x)$, are equal as $x$ approaches the point of interest
  • Example: $\lim_{x \to 0} (-x^2) = \lim_{x \to 0} x^2 = 0$
• Conclude that the limit of the squeezed function, $g(x)$, is equal to the common limit of the squeezing functions
  • Example: By the Squeeze Theorem, $\lim_{x \to 0} x^2 \cos(\frac{1}{x}) = 0$

Limit Problems with Algebraic Techniques

Factoring and Rationalization

• Factoring: Factor the numerator and denominator of a rational function to cancel common factors and simplify the expression before evaluating the limit
  • Example: To find $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$, factor the numerator to get $\lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3}$, cancel the common factor, and evaluate the limit as $\lim_{x \to 3} (x + 3) = 6$
• Rationalization: Multiply the numerator and denominator by the conjugate of the denominator to rationalize the denominator and simplify the expression before evaluating the limit
  • Example: To find $\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x}$, multiply the numerator and denominator by the conjugate of the numerator, $\sqrt{x + 1} + 1$, to get $\lim_{x \to 0} \frac{x}{x(\sqrt{x + 1} + 1)}$, simplify, and evaluate the limit as $\lim_{x \to 0} \frac{1}{\sqrt{x + 1} + 1} = \frac{1}{2}$

Trigonometric Identities, Logarithmic Properties, and Change of Variable

• Trigonometric identities: Use trigonometric identities to simplify expressions involving trigonometric functions before evaluating the limit
  • Example: To find $\lim_{x \to 0} \frac{\sin 2x}{x}$, use the double angle formula $\sin 2x = 2\sin x \cos x$ to get $\lim_{x \to 0} 2\cos x$, and evaluate the limit as $2$
• Logarithmic properties: Use the properties of logarithms to simplify expressions involving logarithmic functions before evaluating the limit
  • Example: To find $\lim_{x \to 1} \frac{\ln x}{\ln(x^3)}$, use the property $\ln(x^3) = 3\ln x$ to get $\lim_{x \to 1} \frac{\ln x}{3\ln x}$, simplify, and evaluate the limit as $\frac{1}{3}$
• Change of variable: Introduce a new variable to simplify the expression or to make the limit more apparent before evaluating the limit
  • Example: To find $\lim_{x \to 0} \frac{e^{3x} - 1}{x}$, let $u = 3x$ and rewrite the limit as $\lim_{u \to 0} \frac{e^u - 1}{u/3}$, which simplifies to $3\lim_{u \to 0} \frac{e^u - 1}{u} = 3$

L'Hôpital's Rule

• L'Hôpital's rule: For indeterminate forms of type $0/0$ or $\infty/\infty$, differentiate the numerator and denominator separately and evaluate the limit of the resulting quotient
  • Example: To find $\lim_{x \to 0} \frac{\sin x}{x}$, which is an indeterminate form of type $0/0$, apply L'Hôpital's rule to get $\lim_{x \to 0} \frac{\cos x}{1}$, and evaluate the limit as $1$
  • Example: To find $\lim_{x \to \infty} \frac{x}{\sqrt{x^2 + 1}}$, which is an indeterminate form of type $\infty/\infty$, apply L'Hôpital's rule to get $\lim_{x \to \infty} \frac{1}{\frac{2x}{2\sqrt{x^2 + 1}}}$, simplify, and evaluate the limit as $1$