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The first concept we will look at is something called a limit. Limits quantify what happens to the values of a function as we approach a given point. This can be defined notationally as:

limx6f(x)=L\lim_{x \rightarrow 6} f(x) = L

We can read this in simple terms as “the limit as x goes to 6 of f(x) approaches some value L”. To evaluate this limit, we take points increasingly closer and closer to 6— as close to 6 as we can get, but not 6 itself! We then evaluate where the function is headed at those points.

If we look at the limit of a function as x approaches a value from one direction, this is called a one-sided limit. For example, we might look at the values of f(5), f(5.9), f(5.999) and see if they are trending towards the value of f(6). This is represented as:

limx6f(x)=L\lim_{x \rightarrow 6^{-}} f(x) = L

We read this as “the limit as x approaches 6 from the left side approaches some value L.”

Whereas if we looked at the values of f(6.1), f(6.01), f(6.0005) and see if they are trending towards f(6), we would represent this as:

limx6+f(x)=L\lim_{x \rightarrow 6^{+}} f(x) = L

We read this as “the limit as x goes to 6 from the right side approaches some value L.”

This takes us to the final key concept about limits. The limit as x approaches 6 exists only if the limit as x approaches 6 from the left side is equal to the limit as x approaches 6 from the right side. This is written out as:

if

limx6f(x)=limx6+f(x)=L \lim_{x \rightarrow 6^{-}} f(x) = \lim_{x \rightarrow 6^{+}} f(x) = L

then

limx6f(x)=L \lim_{x \rightarrow 6} f(x) = L

if

limx6f(x)limx6+f(x) \lim_{x \rightarrow 6^{-}} f(x) \neq \lim_{x \rightarrow 6^{+}} f(x)

then

limx6f(x) does not exist \lim_{x \rightarrow 6} f(x)\ does\ not\ exist

Instructions

The animation to the right shows limit calculations for the function at x=-4 and x=2. In this animation, take note of the following:

  • At x=-4, the limit as x approaches -4 does not exist. This is because the one-sided limits each approach a different value
  • A x=2, the limit as x approaches 2 exists since both one-sided limits approach the same value.

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