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Home / 07 Limits of Functions / 01 Limits

Introduction

To this point, we have not formally introduced the notion of a limit. However, we have used it conceptually quite a bit, specifically, whenever we talk about function end behavior we have in fact, been talking about limits. Let's start graphically and consider a function with both asymptotes and holes and answer the question about various types of limiting behavior we observe in the graph:

f(x) = \frac{\left(x-3\right)\left(x+3\right)^{{2}}}{x\left(x-3\right)\left(x+4\right)}; {hidden:false,color:black,lineWidth:3,secret:true}\ g(x) = \frac{\left(x+3\right)^{{2}}}{x\left(x+4\right)}; {hidden:true,color:black,secret:true}\ (3,g(3));{pointStyle:Desmos.Styles.OPEN,color:black}

In this graph, we will start by focusing on the point value at \(x=3\). Clearly, there is a domain restriction at this value. However, if we were to trace the graph, let's say from left to right, you would observe that the outputs are clearly converging to some numerical value. Similarly, you get the same numerical value if you trace it from right to left. Since both values are the same, it may make sense to declare that the limit as you trace the inputs close to the domain restriction is that numerical value.

Definitions

  1. We say a function \(f(x)\) has a left sided limit as \(x\) approaches \(a\) if the function approaches a constant numerical value \(L\) as you trace the function left to right towards \(a\). We denote this in the following way: \[ \displaystyle\lim_{x\rightarrow a^{-} } f(x)=L\;\;\;\text{Left Sided Limit} \]
  2. We say a function \(f(x)\) has a right sided limit as \(x\) approaches \(a\) if the function approaches a constant numerical value \(L\) as you trace the function right to left towards \(a\). We denote this in the following way: \[ \displaystyle\lim_{x\rightarrow a^{+} } f(x)=L\;\;\;\text{Right Sided Limit} \]
  3. We say a function \(f(x)\) has a limit as \(x\) approaches \(a\) if both the left and right sided limit exist and are the same. We denote this \[ \displaystyle\lim_{x\rightarrow a } f(x)=L\;\;\;\text{Limit of }f(x) \]