Intro to Limits Sections 1.2, 1.3, &1.4

Definition of a Limit If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L. This limit is written as .

How to Find a Limit The easiest way to find a limit is simply to substitute the value that the limit is approaching into your function. Find the value of the following limits at each given value.

You can also use graphs and tables when they are provided. x 2.9 2.99 2.999 3 3.001 3.01 3.1 f(x) 2.710 2.970 2.997 ? 3.003 3.030 3.310

You can still find a limit even if it’s not in the domain. Factor out the numerator and see if anything cancels.

Or you can rationalize a fraction

Non-Existence of a Limit There are 3 things to look for to see if a limit does not exist. The graph might be oscillating at the given point. The graph can be approaching a different number from the left and right. The graph increases or decreases without bound

Oscillating Behavior

Different Behavior from Left and Right

Increases or Decreases without Bound ***In this case we say that . However this does not mean that the limit exists. Instead, this statement shows us how the limit fails.

What to do if there is an absolute value ALWAYS break it apart into a piece-wise function.

One Sided Limits refers to the limit of f(x) as it approaches c ONLY from the left refers to the limit of f(x) as it approaches c ONLY from the right = if and only if exists

Use the graph to find the limits of the greatest integer function given

Use the graph of the piecewise function to find each of the following limits