Finding Limits

[Nicole1234]

I need to know how to find the limit of x to the sixth power minus one divided by x to the tenth minus 1 as x approaches 1. I dont understand how to find limits and the book says the answer is 3/5 i have no clue how to do this.

 

[soroban]

Hello, Nicole1234!

I need to know how to find: \(\displaystyle \L\,\lim_{x\to1}\,\left( \frac{x^6\,-\,1}{x^{10}\,-\,1}\right)\)

You're probably expected to know that both polynomials factor:

\(\displaystyle \;\;x^6\,-\,1\:=\x\,-\,1)(x^5\,+\,x^4\,+\,x^3\,+\,x^2\,+\,x\,+\,1)\)

\(\displaystyle x^{10}\,-\,1\:=\x\,-\,1)(x^9\,+\,x^8\,+\,x^7\,+\,x^6\,+\,x^5\,+\,x^4\,+\,x^3\,+\,x^2\,+\,x\,+\,1)\)

Now try it . . .

 

[Nicole1234]

Replying

I have no idea how to solve limits at all and have no idea how that is supposed to hlep me to solve the problem. I really just need a full explination of the entire way to solve the problem and why each step is being taken i tried to do this and just do not come out with 3/5

 

[stapel]

Apply the suggested factoring, cancel the displayed common factor, and evaluate what is left.

Eliz.