Question

Evaluate the following limit:

Answer 1

\[\lim_{x \rightarrow \infty} \frac{ x^2- x +7}{ 5x^2-6x+12 }\]

Answer 2

I am having a bit a trouble evaluating this one, it is being quite tricky.

Answer 3

^Personifying my limits makes them seem less tricky^

Answer 4

So when you have two of the same exponents in the first part of the limit, you would just take the coefficients out, and that would be the limit. So it would be 1/5.

Answer 5

That is a little confusing, I'm not quite sure I understand

Answer 6

The limit of any polynomial function can be calculated by the term with the highest exponent, as long as the limit tends to infinity, this can be easily proven by taking the limit of a generic polynomial:
\[\lim_{x \rightarrow \infty}a_1x^n+a_2x ^{n-1}+...a _{n-1}x+a_{n}\]
And taking the highest exponent term as common factor:
\[\lim_{x \rightarrow \infty} a _{n}x^n (\frac{ a _{n-1}x^{n-1} }{ a _{n}x^n }...\frac{ a _{2}x^2 }{ a_nx^n } + \frac{ ax }{a_nx^n }+\frac{ a }{ anx^n })\]
Evaluating the corresponding equalities to the infinity we get:
\[\lim_{x \rightarrow \infty}a_nx^n\]
So, it is proven that the limit of a polynomial function as x tends to infinity is equal to the term with the highest exponent.

Moving on the problem in question:
\[\lim_{x \rightarrow \infty}\frac{ x^2-x+7 }{ 5x^2-6x+12 }\]
And applying what I have just proven to you, we get:
\[\lim_{x \rightarrow \infty}\frac{ x^2 }{ 5x^2 }\]
I'll let you take over from here.