Limits Lesson PDF

Okay, here is the transcription of the text in the image, formatted in markdown. ### Lesson: Limits MAPMON **Definition of limits** Let $f$ be a function defined on an open interval containing $c$ (except possibly at $c$) and let $L$ be a real number. The statement $\lim_{x \to c} f(x) = L$ mea...

Okay, here is the transcription of the text in the image, formatted in markdown. ### Lesson: Limits MAPMON **Definition of limits** Let $f$ be a function defined on an open interval containing $c$ (except possibly at $c$) and let $L$ be a real number. The statement $\lim_{x \to c} f(x) = L$ means that for each $\epsilon > 0$ there exists a $\delta > 0$ such that if $0 < |x - c| < \delta $, then $|f(x) - L| < \epsilon$ **Some Basic Limits** Let $b$ and $c$ be real numbers and let $n$ be a positive integer ($n \in \mathbb{Z}$). 1. $\lim_{x \to c} b = b$ 2. $\lim_{x \to c} x = c$ 3. $\lim_{x \to c} x^n = c^n$ **Examples** 1. $\lim_{x \to 2} 3 = 3$ 2. $\lim_{x \to 4} x = -4$ 3. $\lim_{x \to 2} x^2= 2^2 = 4$ **Properties of limits** Let $b$ and $c$ be real numbers, let $n$ be a positive integer, and let $f$ and $g$ be functions with the following limits. $\lim_{x \to c} f(x)$ and $\lim_{x \to c} g(x)$ 1. Scalar multiple: $\lim_{x \to c} b f(x)= b \lim_{x \to c} f(x)$ 2. Sum or difference: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$ 3. Product: $\lim_{x \to c} [f(x) \times g(x)]= \lim_{x \to c} f(x) \times \lim_{x \to c} g(x)$ 4. Quotient: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, $\lim_{x \to c} g(x) \neq 0$ 5. Power: $\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$, where $n \in \mathbb{Z}$ 6. Radical: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)}$ **Examples** $\lim_{x \to a} f(x) = 4$ $ \lim_{x \to a} g(x) = -3$ 1. $\lim_{x \to a} 4 f(x) = 4 \lim_{x \to a} f(x) = 4(4) = 16$ 2. $\lim_{x \to a} [3 f(x) + 5 g(x)] = 3 \lim_{x \to a} f(x) + 5 \lim_{x \to a} g(x) = 3(4) + (5)(-3) = 12 - 15 = -3$ 3. $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = 4(-3) = -12$ 4. $\lim_{x \to a} [g(x)]^4 \cdot \sqrt{f(x)} = [\lim_{x \to a} g(x)]^4 \cdot \sqrt{\lim_{x \to a} f(x)} = (-3)^4 \cdot \sqrt{4} = 81 \cdot (2) = 162$ 5. $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{4}{-3}= -\frac{4}{3}$ **Examples:** 1. $\lim_{x \to 2} (x^2 + x) = \lim_{x \to 2} x^2 + \lim_{x \to 2} x = 2^2 + 2 = 6$ 2. $\lim_{x \to 2} [(x+1)(x-3)] = \lim_{x \to 2} (x+1) \cdot \lim_{x \to 2} (x-3) = (2+1) \cdot (2-3) = 3 \cdot (-1) = -3$ 3. $\lim_{x \to 3} [2(x-1)] = 2\lim_{x \to 3} (x-1)$ or $\lim_{x \to 3} [2x-2] = \lim_{x \to 3} 2x - \lim_{x \to 3} 2 = 2(3) - 2 =4 = 2(3 - 1) = 2 \cdot 2 = 4$ 4. $\lim_{x \to 1} \frac{2x^2 + x - 3}{x^2 + 4} = \frac{\lim_{x \to 1} 2x^2 + x - 3}{\lim_{x \to 1} x^2 + 4} = \frac{2(1)^2 + (1) - 3}{(1)^2 + 4} = \frac{0}{5} = 0$ **Rationalizing Technique (Conjugate method)** **Examples:** 1. $\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x}$ By direct substitution, you obtain the indeterminate form $\frac{0}{0}$ $= \lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x} \cdot \frac{\sqrt{x+1} + 1}{\sqrt{x+1} + 1}$ $= \lim_{x \to 0} \frac{x+1 - 1}{x(\sqrt{x+1} + 1)}$ $= \lim_{x \to 0} \frac{x}{x(\sqrt{x+1} + 1)}$ $= \frac{1}{\sqrt{0+1} + 1} = \frac{1}{2}$ 2. $\lim_{x \to 2} \frac{\sqrt{4x+1} - 3}{x - 2} \qquad + (x-2) \qquad \frac{2}{3}$ 3. $\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{x} \qquad 1$ **Homework** 4. $\lim_{x \to -2} \frac{x+2}{\sqrt{x^2 + 5} - 3} \qquad \frac{3}{2}$ 5. $\lim_{x \to 0} (\frac{1}{x \sqrt{1+x}} - \frac{1}{x}) \qquad \frac{1}{2}$ 6. $\lim_{x \to 0} \frac{\sqrt{x+5} - \sqrt{5}}{x} \qquad \frac{\sqrt{5}}{10}$ **Evaluating limit by factoring** **Examples:** 1. $\lim_{x \to 0} \frac{5x}{x^2 +2x} = \lim_{x \to 0} \frac{5x}{x(x+2)}= \lim_{x \to 0} \frac{5}{x+2} = \frac{5}{0+2}= \frac{5}{2}$ **Homework** 2. $\lim_{x \to 2} \frac{x^2 + 3x - 10}{x - 2} \qquad 7$ 3. $\lim_{x \to 3} \frac{\text{Cim } x^2 - 9}{x - 3} \qquad 8$ 4. $\lim_{x \to 2} \frac{x^3 - 8}{x - 2} \qquad 12$ 5. $\lim_{x \to 4} \frac{4-x}{x^2 - 16} \qquad \frac{-1}{8}$ 6. $\lim_{x \to 4} \frac{x^2 - x - 12}{x^2 - x - 20} \qquad \frac{7}{9}$