How do you find the limit of ((2x^2-6)/(5x-x^2)) as x approaches infinity?
1 Answer
Explanation:
This is one way to approach this problem :
If you notice the highest degree of both the denominator and numerator is 2. We can divide every term by
lim_(x->oo) ((2(cancelx^2)/cancel(x^2) -6/x^2) /((5x)/x^2 -cancel(x^2/x^2)))
lim_(x->oo)(2-6/x^2)/(5/x-1)
lim_(x->oo)(2-6/(oo)^2)/(5/oo-1)
Note: As the denominator get larger, the number will be smaller, and almost close to 0. We can stated as follow
===========================================
The other method is to use L'Hopitals' Rule
If we direct substitute we will get an intermediate form
Direct Sub:
We can simply differentiate numerator and denominator separately like so
Note: Derivative of
Derivative of
5x-x^2 = 5-2x
We can rewrite it as :
by direct substitution, we get
We can differentiate again to get