Introduction to Limits

image of traffic light Limits are a part of our daily life. When we think about how fast we can brake for a red light or how much weight we can lift or how long a battery will last, limits are involved.

An Informal Look

The idea of a limit can be related to the input/output model for functions. As inputs to a function f approach a number a, the outputs from f approach a value L. View the following informal introduction to limits presentation.

Left and Right Approaches

Approaching a number is accomplished by starting at values either below (less than) or above (greater than) the number. In limit language approaching the number from below is referred to as a left-hand limit and approaching the number from above is referred to as a right-hand limit. Each of these limits is described as a one-sided limit.

image of left hand A left-hand limit is written as $LaTeX: \lim_{x\to a-}f\left(x\right)=L$ and means that f(x) approaches L as x approaches a from the left. An alternative notation is $LaTeX: f\left(x\right)\rightarrow L$ as equation image indicator $LaTeX: x\rightarrow a$ from the left. The negative sign as an exponent (a^- ) denotes a left approach.

image of right hand A right-hand limit is written $LaTeX: \lim_{x\to a+}f\left(x\right)=L$ equation image indicator and means that f(x) approaches L as x approaches a from the right. An alternative notation is $LaTeX: f(x)\to L$ as $LaTeX: x \to a$ from the right. The positive sign as an exponent (a⁺) denotes a right approach. View the presentation below to see examples of left-hand and right-hand limits.

Informal Definition

Suppose that f is a function defined near, but not necessarily at, x = a. Then the limit of a function f, equation image indicator $LaTeX: \lim_{x\to a}f\left(x\right)=L$ , means that f(x) approaches L as x approaches a. This limit $LaTeX: \lim_{x\to a}f\left(x\right)$ exists if and only if both corresponding one-sided limits exist and are equal. Symbolically, this is written as $LaTeX: \lim_{x\to a}f\left(x\right)=L\Leftrightarrow \lim_{x\to a+}f\left(x\right)=L\lim_{x\to a-}f\left(x\right)$

Introduction to Limits: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of one-sided and two-sided limits.

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