Question

In calculus if you are finding the limit of something as it approaches infinity and it's just a line, what are you supposed to say?

Answer 1

My teacher said not to write it approaches infinity

Answer 2

That depends on what the line approaches at infinity. If the line is increasing, then it approaches positive infinity. If it's decreasing, it approaches negative infinity. If it's constant, it approaches that constant, and if it's vertical it's undefined at infinity.

Answer 3

what kind of line?

Answer 4

Like 2x+5

Answer 5

well the limit as x goes to infinity of that is +infinity, but you can also say DNE for "does not exist"

Answer 6

...since infinity is not technically a number

Answer 7

okay so if it goes to infinity, the limit does not exist?

Answer 8

\[\lim_{x\to\infty}(2x+5)=+\infty\] For this to be entirely accurate, you have to be working in the extended reals, where \(+\infty\) is actually defined.

Answer 9

It kind of depends on what your teacher is looking for. There is a fair argument that the limit is not defined, because what would the delta and epsilon be?

Answer 10

yeah, limits that tend to +/-infinity are t3echnically nonexistent (because they never settle on a single value)

I actually learned that here :)

Answer 11

technically*

Answer 12

but there are different kinds of non-existent limits, like in the way that sinx has no limit as x goes to infinity (since sin oscillates between -1 and 1 forever)

saying that the limit is infinity is just a more specific kind of DNE