Understanding Limits: Cheat Sheet

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Essential Concepts

A Preview of Calculus

Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative. Calculating a derivative requires finding a limit.
Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the [latex]x[/latex]-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.

Introduction to the limit of a function

A table of values or graph may be used to estimate a limit.
If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
We may use limits to describe infinite behavior of a function at a point.

Slope of a Secant Line
[latex]m_{\sec}=\dfrac{f(x)-f(a)}{x-a}[/latex]
Average Velocity over Interval [latex][a,t][/latex]
[latex]v_{\text{avg}}=\dfrac{s(t)-s(a)}{t-a}[/latex]
One-Sided Limits
[latex]\underset{x\to a^-}{\lim}f(x)=L[/latex]
[latex]\underset{x\to a^+}{\lim}f(x)=L[/latex]
Intuitive Definition of the Limit
[latex]\underset{x\to a}{\lim}f(x)=L[/latex]

infinite limit: A function has an infinite limit at a point [latex]a[/latex] if it either increases or decreases without bound as it approaches [latex]a[/latex]

intuitive definition of the limit: If all values of the function [latex]f(x)[/latex] approach the real number [latex]L[/latex] as the values of [latex]x(\ne a)[/latex] approach [latex]a[/latex], [latex]f(x)[/latex] approaches [latex]L[/latex]

one-sided limit: A one-sided limit of a function is a limit taken from either the left or the right

vertical asymptote: A function has a vertical asymptote at [latex]x=a[/latex] if the limit as [latex]x[/latex] approaches [latex]a[/latex] from the right or left is infinite

The Tangent Problem and Differential Calculus

Practice sketching secant lines for various functions and intervals.
Visualize how secant lines approach the tangent line as points get closer.
When working with position functions, relate concepts of average and instantaneous velocity to secant and tangent lines.
Practice calculating average velocities over smaller and smaller intervals.

The Area Problem and Integral Calculus

Practice approximating areas using different numbers of rectangles. Understand the relationship between smaller rectangles and better approximations.
Visualize how the sum of rectangle areas approaches the true area as widths decrease.
Practice setting up integrals for various shaped regions.

The Definition of a Limit

Practice estimating limits using both tables and graphs.
Be aware that a function can be defined at a point but have a different limit there.
Look for oscillating behavior when suspecting a limit might not exist.
Remember that limits describe behavior near a point, not necessarily at the point.
Use basic limit properties to simplify complex limit problems.

One-Sided and Two-Sided Limits

When evaluating limits, always consider both left-hand and right-hand approaches.
For piecewise functions, use the appropriate piece for each one-sided limit.
Create tables of values approaching the point of interest from both sides.
Remember that a function can be continuous from one side but not the other.
When solving limit problems, explicitly state both one-sided limits before concluding about the two-sided limit.

Infinite Limits

Remember that infinite limits are not actual limits but descriptions of behavior.
When finding vertical asymptotes, check both sides of the potential asymptote.
Use factoring to identify potential vertical asymptotes in rational functions.
Distinguish between vertical asymptotes and holes in rational functions.