Precise Definition of a Limit ε, δ

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Precise Definition of a Limit ε, δ

The limit of f of x as x goes to a

$\lim_{x \to a} f(x) size=large$

f of x approaches the limit as x approaches a

$f(x) \to L \text{ as } x \to a size=large$

Limit

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

Left Hand Limit

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

Right Hand Limit

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

Derivation of “The Difference Quotient”

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

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

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

Slope of Secant Line or Difference Quotient

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

Intermediate Value Theorem

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

(∃ means “such that”)

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Precise Definition of a Limit ε, δ

Limit

Left Hand Limit

Right Hand Limit

Derivation of “The Difference Quotient”

Slope of Secant Line or Difference Quotient

Intermediate Value Theorem

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