Precise Definition of a Limit ε, δ

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The limit of f of x as x goes to a

$\Large \lim_{x \to a} f(x)$

f of x approaches the limit as x approaches a

$\Large f(x) \to L \text{ as } x \to a$

For every $\epsilon > 0$ , there is a $\delta > 0$ such that $latex 0 < |x - a| < \delta$ and $latex |f(x) - L| < \epsilon$

For every $\epsilon > 0$ , there is a $\delta > 0$ such that $latex a – \delta < x < a$ and $latex |f(x) - L| < \epsilon$

For every $\epsilon > 0$ , there is a $\delta > 0$ such that $latex a < x < a + \delta$ and $latex |f(x) - L| < \epsilon$

Derivation of “The Difference Quotient”

$\Large m = \frac{\Delta y}{\Delta x} \Rightarrow \frac{\Delta y}{\Delta x} = \frac{y - y_0}{x - x_0}, \quad y = f(x)$

$\Large \Rightarrow \frac{y - y_0}{x - x_0} = \frac{f(x) - f(x_0)}{x - x_0}, \quad \Delta x = x - x_0 \Leftrightarrow x = \Delta x + x_0$

$\Large \Rightarrow \frac{f(x) - f(x_0)}{x - x_0} = \frac{f(\Delta x + x_0) - f(x_0)}{\Delta x} \equiv \frac{f(x + h) - f(x)}{h}$

$\Large m = \frac{f(x + h) - f(x)}{h} = \frac{f(x + \Delta x) - f(x)}{\Delta x} \Leftrightarrow h = \Delta x$

If $f$ is continuous on $[a,b]$ , $latex f(a) < N < f(b)$ and $latex f(a) \ne f(b)$, then there is a $latex c \in (a,b)$ $latex \exists f(c) = N$

(∃ means “such that”)

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