Sketch the graph of the function and use it to determine the...

Sketch the graph of the function and use it to determine the values of $a$ for which $\lim _{x \rightarrow a} f(x)$ exists.
$$
f(x)=\left\{\begin{array}{ll}
e^{x} & \text { if } x \leqslant 0 \\
x-1 & \text { if } 0<x<1 \\
\ln x & \text { if } x \geqslant 1
\end{array}\right.
$$

Key Concepts

One-Sided Limits

One-sided limits involve calculating the limit of a function as the input approaches a point exclusively from one side, either from the left or from the right. In the context of piecewise functions, evaluating these limits at the boundary points helps to determine if the two-sided limit exists.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of its domain. Understanding such functions involves analyzing each piece on its defined interval as well as the transitions between these intervals, particularly at the boundaries where the rule changes.

Limit Existence

The existence of a limit at a point requires that the left-hand limit and the right-hand limit exist and are equal. This concept is fundamental in ensuring that as the input approaches the point from either side, the output approaches a single, specific value.

Continuity

Continuity of a function at a point means that the function is defined at that point, and the limit as the input approaches that point equals the function's value. When dealing with piecewise functions, checking continuity at the boundaries where the function's definition changes is critical for determining the overall behavior of the function.

Graphical Analysis

Graphical analysis involves sketching or visualizing the function to identify key features such as trends, continuity, and points of discontinuity. Graphs are particularly useful in understanding the behavior of piecewise functions and verifying limit existence by observing the behavior near the boundaries.

Transcript

00:01 Okay, this is the problem we're working on.

00:03 Sketch the graph of the function and use it to determine the values for a, of a, for which the limit as x approaches a of f of x exists.

00:14 Just real quick, since we think about this, this looks like a weird function because it's a piecewise function, but in general for a function to have a limit exist, it has to be continuous and has, and there has to be no holes or gaps.

00:30 And actually you can have a hole, it just depends on the graph and what it's looking like.

00:36 So let's look at these, and notice they give us three different functions.

00:42 They give us e to the x, they give us x minus one and natural log of x, and we're just working here, e to the x is going to be all the x values less than zero, we're going to graph x minus one in between zero and one, and the natural log of x, and greater than one, so i've got this all sketched up right here.

01:08 Notice over here on the left are the functions.

01:11 The red function is e to the x, that looks pretty familiar.

01:15 I've got a, i've got a purple vertical line at x equals zero, because that's the boundary where we stop worrying about e to the x, we start focusing on x minus one, which is down here, this blue one.

01:30 And then i have a black vertical line at x equals one, and that's where we stop worrying about x minus one, and we start worrying about the natural log of x, which is the green graph here.

01:45 So if we're coming in from negative, from negative infinity, and we're looking, this function is continuous all the way into zero...

Please give Ace some feedback