When looking at a fraction of a limit to infinity, you need to only look at the highest power terms of the denominator and numerator.
I'm in a beginning calculus class after not touching math of any sort for 4 years. I've forgotten a lot of the more complicated algebra, and I'm struggling with simplifying limits right now. I'm doing my homework right now and coming across a lot of problems I'm getting wrong and I can't understand why, so I am asking for help.
first problem:
1.)
To work this one out, I multiplied the top and bottom by ,
so I get
From there, I divided everything by since that's the highest x value in the denominator in order to get rid of some of the numbers. I wind up with:
Which is which equals
The back of my book however says this answer should just be 3. Why is the x^3 cancelled?
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Here's another I got wrong:
All I could think to do was to divide everything by X but I don't even think you're allowed to do that without a denominator (are you?). Anyway when I do that I get
Apparently the answer is supposed to be 1/6, how do you get this?
L'Hospital's Rule works nicely in both cases. If you have an indeterminate form or , then
The first is already in form, and the second can be made into that form using this transformation...
Recall that because of the continuity of the logarithm, , and the "stuff" inside the logarithm is now of the form . So you can use L'Hospital's Rule there too
Thanks for all that help, but I haven't learned this rule yet and I don't feel comfortable enough trying stuff outside of what I've already been taught. Thanks for all that effort though!
DWsmith:
Thanks for the first reply, I realized i made a stupid mistake on the first problem i posted and after working that out I got 3. Conjugating first was all I needed for the second problem as well.
Another question, How do you find horizontal and vertical asymptotes? I know for horizontal you have to make the limit of a function + or - infinity, and I'd assume for vertical it's the limit of a function when x approaches 0? I can provide a practice problem if it would be easier to grasp what im saying.
Vertical asymptote occur when the denominator is zero.
Horizontal--you need to check the limits at negative and positive infinity.
Slant--when the highest degree of the numerator is greater than the highest degree of the denominator. To solve, you do long division and disregard the remainder.