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2-0 Limits

Learning Targets

You will be able to

Concepts / Definitions

Informal definition of a limit

If $f(x)$ becomes arbitratily close to a single number $L$ as $x$ approaches a number $a$ from either side, the limit of $f(x)$ as $x$ approaches $$ is $L$.

Limit Definition

Definition for a Limit using one-sided limits

\(\lim_{x\to a} f(x) = L\quad iff\quad \lim_{x\to a^-} f(x) = L = \lim_{x \to a^+} f(x)\)

Note: Technically, the theorem is based on a real number, $L$, but we include infinity as a possible answer.

Input x approaches a

Graphical Example

Graphical Example

Numerical Example

Numerical Example

Numerically, we will have a limit if it is clear we are approaching one output value as we look at values from inputs smaller and larger than $\alpha$.

Analytic / Algebraic Example

\(\lim_{x\to 1}\frac{x^2-1}{x-1}\) \(= \lim_{x\to 1}\frac{(x+1)(x-1)}{x-1}\) \(= \lim_{x\to 1}(x+1)\) \(= 2\)

Many times limits (not all), like the derivative, require us to find the output value of the hole of the difference quotient.

Continuity at a Point

Interior Point

A function $y = f(x)$ is continuous at an interior point c of its domain if \(\lim_{x\to c} f(x) = f(c)\)

Endpoint

A function $y = f(x)$ is continuous at a left endpoint $\alpha$ or is continuous at a right endpoint $b$ of its domain if \(\lim_{x\to a^+} f(x) = f(a)\quad or\quad \lim_{x\to b^-} f(x) = f(b)\ respectively\)

Intermediate Value Theorem for Continuous Functions

A function $y = f(x)$ that is continuous on a closed interval $[a, b]$ takes on every value between $f(a)$ and $f(b)$. In other words, if $y_0$ is between $f(a)$ and $f(b)$, then $y_0 = f(c)$ for some $c$ in $[a, b]$.

Intermediate Value Theorem

Properties of Limits

Let $b$ and $c$ be real numbers, lef $n$ be a positive integer, and let $f$ and $g$ be functions with the following limits.

Properties of Limits

Ex: Is any real number exactly 1 less than its cube?