Limits

Reflection 12 would go here

How to write asymptotes in limit notation

Source:

Explanation:

This artifact demonstrates how to write asymptotes in limit notation.

Example 1)

As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large.

The limit of f(x) as x approaches infinity is 2.

Example 2)

As x approaches 0, \(f(x)\) approaches positive or negative infinity, depending on which direction x is approaching from. This is because the closer the number gets to 0, the smaller it has got to be. For x-values like 0.0001, \(f(x)\) will actually be a big number because \(f(x) = {1 \over x}\)

Inversely, as x approaches positive or negative 0, \(f(x)\) approaches 0 because \(f(x)\) is always a fraction. As the demominator of the fraction increases, it’s value decreases and is getting closer and closer to 0.

Artifact:

Example 1)

\(f(x) = {2x - 1 \over x}\\ \lim_{x \to \infty} {2x - 1 \over x} = 2\)

Example 2)

\(f(x) = {1 \over x}\\ \lim_{x \to 0} {1 \over x} = \pm \infty\\ \lim_{x \to \pm \infty} {1 \over x} = 0\)

../_images/12_asymptotes.png

Removable discontinuity

Source: Online Tutorial

Awareness and Appreciation:

I didn’t know how to do this, but Igoogled it succesfully.

This demonstrates that I am aware of the magnificient resources that are available to me, and that I appreciate them because I use them.

Independent Thinking

I didn’t know what to show about removable discontinuity, I just knew what it was.

So I went online and read about it to figure out how it works.

Explanation:

This artifact demonstrates removable discontinuity.

Discontinuity is removable if you can easily plug in the holes in its graph by redefining the function.

In the original function, for -2 and 2 f(x) is undefined. But a little bit of algebraic magic reveals that the function can be “patched” for f(2) to be successful.

Artifact:

\[f(x) = {x-2 \over x^2 - 4}\]\[{x - 2 \over (x-2) (x+2)}\]\[{1 \over x+2} \text{ The (x-2)s cancel out}\]\[f(2) = {1 \over 4}\]\[f(x) = {1 \over 4} \text{ if x = 2}\]\[x = -2 \text{ Vertical Asymptote}\]\[x = 2 \text{ Removable Discontinuity}\]