Indeterminate forms are tricky expressions that pop up when evaluating limits. They're like mathematical puzzles that need extra steps to solve. There are seven common types, including the infamous 0/0 and ∞/∞.

To crack these puzzles, we use clever tricks like factoring, canceling, and applying L'Hôpital's Rule. We also analyze how functions behave near tricky points and use the Squeeze Theorem to pin down elusive limits.

Indeterminate Forms

Seven indeterminate forms

• Expressions that cannot be immediately evaluated to a definite value arise when evaluating limits and direct substitution fails ($\frac{0}{0}$, $\frac{1}{0}$)
• Seven common indeterminate forms: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $1^\infty$, $\infty - \infty$, $0^0$, $\infty^0$
• Each form represents a situation where the limit cannot be determined without further analysis ($\lim_{x \to 0} \frac{\sin x}{x}$, $\lim_{x \to \infty} (1 + \frac{1}{x})^x$)

Evaluation of indeterminate limits

• Manipulate indeterminate expressions using algebraic techniques to evaluate the limit ($\lim_{x \to 0} \frac{x^2 - 1}{x - 1}$)
• Factor and cancel common factors, multiply by conjugates, rationalize numerators or denominators, apply trigonometric identities ($\lim_{x \to 0} \frac{\sin 3x}{\sin 5x}$)
• Simplify expressions involving exponential and logarithmic functions using their properties ($\lim_{x \to \infty} \frac{\ln x}{x}$)
• Apply L'Hôpital's Rule for $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms: the limit of a quotient equals the limit of the quotient of their derivatives, if it exists ($\lim_{x \to 0} \frac{e^x - 1}{x}$)

Squeeze theorem for limits

• Finds the limit of a function "squeezed" between two other functions with known limits ($\lim_{x \to 0} x^2 \sin(\frac{1}{x})$)
• If $f(x) \leq g(x) \leq h(x)$ near $a$ (except possibly at $a$), and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$
• Find two functions $f(x)$ and $h(x)$ with the same limit $L$ as $x \to a$, show $g(x)$ is always between them near $a$, conclude $\lim_{x \to a} g(x) = L$ ($0 \leq 1 - \cos x \leq \frac{x^2}{2}$ as $x \to 0$)

Function behavior near indeterminate points

• Analyze the function's behavior in the neighborhood of a point where an indeterminate form occurs ($\lim_{x \to 0^+} \frac{1}{x}$, $\lim_{x \to 0^-} \frac{1}{x}$)
• Evaluate one-sided limits to check if the function approaches the same value from both sides; equal one-sided limits imply the limit exists ($\lim_{x \to 0^+} \frac{1}{\sqrt{x}}$, $\lim_{x \to 0^-} \frac{1}{\sqrt{x}}$)
• Investigate the function's behavior as $x$ approaches the point from the left and right; determine if it increases without bound, decreases without bound, or oscillates ($\lim_{x \to 0} \frac{1}{\sin x}$)
• Sketch the function near the point to visualize its behavior; identify vertical asymptotes, horizontal asymptotes, or holes in the graph ($y = \frac{1}{x^2}$, $y = \frac{x^2 - 1}{x - 1}$)