Take It To The Limit by Roger Pink

Limits!

A limit is the value that a function f(x) approaches as the input (x) of that function approaches a value. Sometimes it is useful to take the "limit at infinity". This is basically saying, as x gets larger and larger, does the function converge upon a certain value (0, 15, 972, 1/3, 2.71828..., etc.) or does the function diverge (∞,-∞, etc.)?

Let's look at an example to see how this works:

So the above says "as x gets larger and larger and larger, the function 1/x approaches 0". Does that make sense?

1/10, 1/100, 1/1000, 1/10000,.... Yup, seems to be heading towards zero. Let's look at another example.

Ok, sure. We care about where it's going, not how fast it's getting there. This function is definitely approaching zero as x gets larger. Getting there faster than the last function (1/x) too.

Ok, so this is less clear. Certainly written this way the answer isn't obvious. Best to rewrite it using some algebra.

And that is how it's done. Use some algebra to reduce it to a form that is clear to solve. Here are some examples:

Sometimes the limit doesn't converge. Here's an example:

Makes sense. Certainly as x gets larger and larger, x gets larger and larger. Please remember that infinity is not a number, but an idea. It basically means "continues on". So the limit of a function that equals infinity is not converging at a number but diverging (continuing on). Here's some more examples:

Here's a few more examples of limits of functions that converge. Again, these need to be modified algebraically in order to clearly see the limits they converge to.

If you are confused as to how any of the above were solved, please check out this great link that will explain in detail the algebraic steps taken to find the limit.

U C Davis Math ~ Limits

Astronomy Fans!

Some cool news regarding our local Galactic Neighborhood. Find the article here.

It reads:

We live in a galaxy known as the Milky Way -- a vast conglomeration of 300 billion stars, planets whizzing around them, and clouds of gas and dust floating in between. Though it has long been known that the Milky Way and its orbiting companion Andromeda are the dominant members of a small group of galaxies, the Local Group, which is about 3 million light years across, much less was known about our immediate neighborhood in the universe. Now, a new article maps out bright galaxies within 35-million light years of the Earth, offering up an expanded picture of what lies beyond our doorstep. Cont. Here

If you haven't had a chance, check out "You Are Here" found on my website (Roger H Pink) which explains where we are in the universe at various scales.

As always, thanks for stopping by and reading my blog. - Roger