Easy yet tricky limit question (showing my work)

I have to show my work on a problem:

Ok, so first thing: divide by fastest growing term in the denominator, which in this case as x approaches is actually:

So: =

This is where I start to get a little confused. How exactly do I mathematically express that as x approaches approaches zero?

I can say: , but then what? I have seen direct substitution with but isn't this bad form? You can't do arithmetic on infinity.

So then does it just "jump" to: ?

Or is there some merit to doing this next (after splitting the func. to a quotient of limits): ? I thought this had no meaning; if it does what is the next step?

Thank you for any help, I know I got the answer, but it's the showing of the steps I am a little tripped up on. IF there is anything missing, can you also explain then how I can exactly express the other horizontal limit ?

Re: Easy yet tricky limit question (showing my work)

Quote:

Originally Posted by

**rortiz81** I have to show my work on a problem:

Ok, so first thing: divide by fastest growing term in the denominator, which in this case as x approaches

is actually:

So:

=

This is where I start to get a little confused. How exactly do I mathematically express that as x approaches

approaches zero?

I can say:

, but then what? I have seen direct substitution with

but isn't this bad form? You can't do arithmetic on infinity.

No, you shouldn't just replace x with "infinity" but you can use the basis limit "properties":

And now you can replace those **limits** with their values to get .

Quote:

So then does it just "jump" to:

?

Or is there some merit to doing this next (after splitting the func. to a quotient of limits):

? I thought this had no meaning; if it does what is the next step?

Thank you for any help, I know I got the answer, but it's the showing of the steps I am a little tripped up on. IF there is anything missing, can you also explain then how I can exactly express the other horizontal limit

?

Re: Easy yet tricky limit question (showing my work)

Thank you, I skipped the sum/difference of limits law in both the numerator and denominator for brevity, but I will include that in my work.