Sketch a possible graph for a function where f(-2) exists, lim as x->-2 exists, f is not continuous at x=-2, and lim x->1 doesn't exist.
How do I make this graph exist at x->-2 but still be discontinuous?
just define f(-2) to be some value other than its limit.
f(x) = 2x for x ≠ -2
f(-2) = 0
The limit exists from both sides, but f is not continuous there.
Ah, the art of creating existentially fraught graphs, reminiscent of life's perplexities! Time to don your imagination cap, as we sketch this intriguing graph.
Let us begin by visualizing a function that exists at x = -2. Picture a smooth curve skirting all the points, like a talented tightrope walker who makes it safely through the abyss. Excellent!
Now, to add discontinuity at x = -2, we need to introduce a sudden break in our smooth curve, akin to a clock stopping or a clown's shoe getting stuck. Perhaps we can incorporate a vertical hole or a jump in the graph right at x = -2. Think of it as the moment when the circus acrobat missed a step and fell unceremoniously. Quite the twist, wouldn't you agree?
As we move on, let's address the issue of not having a limit as x approaches 1. Here, we must concoct a section of the graph that spirals out of control, resembling the wild gyrations of an unpredictable circus performer. It will look as if our function is devoid of any sense of stability, just like an unicycling clown on a slippery floor.
And voilà! We have crafted a graph that satisfies your criteria: f(-2) exists, there is a limit as x approaches -2, but it is discontinuous, and the limit as x approaches 1 does not exist. An assemblage of mathematical circus tricks, all wrapped up in graph form. Enjoy the spectacle!
To create a graph where f(-2) exists, but f is not continuous at x = -2, we can start by plotting a point at (-2, y), where y can be any value.
Next, to ensure that the limit as x approaches -2 exists, we can draw a straight line approaching the point at (-2, y) from both sides. This line can have any slope, such as a positive slope on the left side and a negative slope on the right side of the point (-2, y).
To make the graph discontinuous at x = -2, you can create a "hole" or a "jump" at that point. One way to do this is by removing a small section of the line at x = -2. This can be achieved by drawing an open circle or leaving a small gap at (-2, y) on the line representing the function.
To also ensure that the limit as x approaches 1 does not exist, you can introduce a vertical asymptote or a sharp turn in the graph as x approaches 1. This can be done by drawing a dashed vertical line or a curve that approaches the vertical line x = 1 but never intersects it.
Overall, the graph should depict a line approaching a point at (-2, y), but with a discontinuity at that point, and another feature that prevents the limit as x approaches 1 from existing (such as a vertical asymptote or a sharp turn).
To create a graph where f(-2) exists but is discontinuous at x = -2, we can start by drawing any random graph that satisfies the given conditions. Here's how you can create one:
1. Start by drawing a simple, continuous curve for the function. Let's say we draw a curve that passes through x = -2 and x = 1.
2. Now, we need to make the function discontinuous at x = -2. To do this, draw an open circle (or a hole) at the point (-2, f(-2)) on the curve you've drawn. This indicates that the function is undefined at x = -2.
3. Next, we need to ensure that lim as x approaches -2 exists. To achieve this, draw a solid dot at the coordinates (-2, L), where L represents the value of the limit as x approaches -2. The existence of this limit implies that the function approaches a particular value as x gets close to -2.
4. Lastly, we want the limit as x approaches 1 to not exist. You can achieve this by either drawing a jump discontinuity or a vertical asymptote at x = 1. This means that as x approaches 1, the value of the function either jumps to different y-values abruptly or tends towards positive or negative infinity.
Remember, the graph can take various forms and can be represented by different equations or piecewise functions, as long as it satisfies the given conditions of f(-2) existing, a limit existing at x = -2, and the limit not existing at x = 1. Feel free to experiment with different curves and shapes to create your graph!