Limits Cheatsheet

Cheatsheet Content

Definition of a Limit Informal: The limit of $f(x)$ as $x$ approaches $c$ is $L$ if $f(x)$ gets arbitrarily close to $L$ as $x$ gets arbitrarily close to $c$ (from either side), but not necessarily equal to $c$. Notation: $\lim_{x \to c} f(x) = L$. Formal ($\epsilon-\delta$ definition): For every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 One-Sided Limits Limit from the right: $\lim_{x \to c^+} f(x) = L$ (as $x$ approaches $c$ from values greater than $c$). Limit from the left: $\lim_{x \to c^-} f(x) = L$ (as $x$ approaches $c$ from values less than $c$). Existence of a limit: $\lim_{x \to c} f(x) = L$ if and only if $\lim_{x \to c^-} f(x) = L$ and $\lim_{x \to c^+} f(x) = L$. Properties of Limits Assume $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$. Let $k$ be a constant. Scalar Multiple: $\lim_{x \to c} [k \cdot f(x)] = k \cdot L$ Sum/Difference: $\lim_{x \to c} [f(x) \pm g(x)] = L \pm M$ Product: $\lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M$ Quotient: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \ne 0$. Power: $\lim_{x \to c} [f(x)]^n = L^n$, for any positive integer $n$. Root: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$, provided $L \ge 0$ if $n$ is even. Composition: If $\lim_{x \to c} g(x) = L$ and $\lim_{y \to L} f(y) = f(L)$, then $\lim_{x \to c} f(g(x)) = f(L)$. Techniques for Evaluating Limits 1. Direct Substitution If $f(x)$ is a polynomial, rational function (where denominator is non-zero), trigonometric, exponential, or logarithmic function, and $c$ is in its domain, then $\lim_{x \to c} f(x) = f(c)$. Example: $\lim_{x \to 2} (x^2 + 3x - 1) = 2^2 + 3(2) - 1 = 4 + 6 - 1 = 9$. 2. Factoring and Canceling Used when direct substitution yields the indeterminate form $\frac{0}{0}$. Factor the numerator and denominator, then cancel common factors. Example: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x-3)(x+3)}{x-3} = \lim_{x \to 3} (x+3) = 3+3=6$. 3. Rationalizing Used when expressions involve square roots and yield $\frac{0}{0}$. Multiply the numerator and denominator by the conjugate of the term containing the root. Example: $\lim_{x \to 0} \frac{\sqrt{x+1}-1}{x} = \lim_{x \to 0} \frac{(\sqrt{x+1}-1)(\sqrt{x+1}+1)}{x(\sqrt{x+1}+1)} = \lim_{x \to 0} \frac{(x+1)-1}{x(\sqrt{x+1}+1)} = \lim_{x \to 0} \frac{x}{x(\sqrt{x+1}+1)} = \lim_{x \to 0} \frac{1}{\sqrt{x+1}+1} = \frac{1}{\sqrt{0+1}+1} = \frac{1}{2}$. 4. Using Common Denominators Used for expressions with fractions that yield $\frac{0}{0}$. Combine fractions in the numerator or denominator. Example: $\lim_{x \to 0} \frac{\frac{1}{x+2}-\frac{1}{2}}{x} = \lim_{x \to 0} \frac{\frac{2-(x+2)}{2(x+2)}}{x} = \lim_{x \to 0} \frac{-x}{2x(x+2)} = \lim_{x \to 0} \frac{-1}{2(x+2)} = \frac{-1}{2(2)} = -\frac{1}{4}$. Limits Involving Infinity 1. Limits at Infinity (Horizontal Asymptotes) $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$. For rational functions $f(x) = \frac{P(x)}{Q(x)}$: If degree of $P(x)$ If degree of $P(x)$ = degree of $Q(x)$, then limit is ratio of leading coefficients. If degree of $P(x)$ > degree of $Q(x)$, then limit is $\infty$ or $-\infty$ (no horizontal asymptote). Key limits: $\lim_{x \to \pm \infty} \frac{1}{x^n} = 0$ for $n > 0$. 2. Infinite Limits (Vertical Asymptotes) Occur when $f(x)$ approaches $\pm \infty$ as $x$ approaches $c$. Typically happens when the denominator of a rational function approaches $0$ and the numerator does not. Example: $\lim_{x \to 0^+} \frac{1}{x} = \infty$, $\lim_{x \to 0^-} \frac{1}{x} = -\infty$. Therefore, $\lim_{x \to 0} \frac{1}{x}$ does not exist. Special Limits $\lim_{x \to 0} \frac{\sin x}{x} = 1$ $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ $\lim_{x \to 0} (1 + x)^{1/x} = e$ $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$ Squeeze Theorem (Sandwich Theorem) If $h(x) \le f(x) \le g(x)$ for all $x$ in an open interval containing $c$ (except possibly at $c$ itself), and if $\lim_{x \to c} h(x) = L$ and $\lim_{x \to c} g(x) = L$, then $\lim_{x \to c} f(x) = L$. Useful for oscillating functions like $\sin(1/x)$. Example: $\lim_{x \to 0} x^2 \sin(\frac{1}{x})$. Since $-1 \le \sin(\frac{1}{x}) \le 1$, we have $-x^2 \le x^2 \sin(\frac{1}{x}) \le x^2$. As $\lim_{x \to 0} (-x^2) = 0$ and $\lim_{x \to 0} x^2 = 0$, by the Squeeze Theorem, $\lim_{x \to 0} x^2 \sin(\frac{1}{x}) = 0$. Continuity A function $f$ is continuous at a point $c$ if all three conditions are met: $f(c)$ is defined. $\lim_{x \to c} f(x)$ exists. $\lim_{x \to c} f(x) = f(c)$. Types of Discontinuities: Removable: The limit exists, but $f(c)$ is undefined or $f(c) \ne \lim_{x \to c} f(x)$. (e.g., hole in the graph) Jump: Left and right limits exist but are not equal. (e.g., piecewise functions) Infinite: One or both one-sided limits are $\pm \infty$. (e.g., vertical asymptotes) Intermediate Value Theorem (IVT) If $f$ is continuous on the closed interval $[a, b]$, and $k$ is any number between $f(a)$ and $f(b)$ ($f(a) \ne f(b)$), then there exists at least one number $c$ in $(a, b)$ such that $f(c) = k$. Useful for proving the existence of roots or solutions to equations.