Question

What does the + and the - mean next to the 1 in x->1? whats the difference between the two?

Answer 1

In this question, we will understand which is the significance of lateral limits, as the following

`\begin{gathered} \lim_(x\rightarrow1^+)f(x_)=2 \\ \\ \lim_(x\rightarrow1^-)f(x)=2 \end{gathered}`

The sign as exponent in the limit means that it is a lateral limit, and has the following graphical interpretation

The right lateral limit

The figure above represent the right lateral limit, it means that we will get closer and closer to the value of the function at x=1, but only approximating by points that are on the right of the point x=1, for example x=1.01, x=1.001, etc...

The limit process in this lateral limit is carry, only on one side of the domain (the right side in this case) , and not on both sides.

The left lateral limit

In the other hand, the left lateral limit is the process of getting closer and closer to the value of the function at x=1, but only approximating by points that are on the left of the point x=1, for example x=0.9, x=0.99,x=0.999, etc. Let us illustrate it with a draw

Finally, let us point that the right limit and the left limit, don't need to match, consider as example the following draw

In this situation,the behavior of the function near of the point x=1, depends on the side that we choose to approximate,

In one hand, if we arrive by the right side of the point x=1, we will see that the function approximate the value y=1. In the other hand, if we chose to arrive by the left side of the value x=1, we will see that the function approximate the value y=-1

Finally, let us point that in the case that the lateral limits, exist and coincides, then we say that the bilateral limit (the normal limit ) exists and is equal to any of the lateral limits.