Need your help on a Limit problem below.?

Question

lim . .. .. .. ..... .... (1 - 4n)/(1 - 2n) =
n=>oo

Answer 1

smci is correct. simply multiply the numerator and demonator by -1/n and get (4-1/n) / (2 -1/n), take the limits which gives
(4-0)/(2-0) or 2. using 1/infinity = 0.

Answer 2

The least complicated and most standard method is to divide numerator and denominator by the largest possible power of n (in this case simply n itself).
lim n→∞ (1 - 4n) / (1 - 2n)
= lim n→∞ (1/n - 4) / (1/n - 2)
= lim n→∞ (4 - 1/n) / (2 - 1/n) ; (negate both terms)
= 4 / 2
= 2

You can see this agrees with the synthetic division.

Answer 3

lim (1 - 4n) / (1 - 2n)
n -> infinity

To solve this without using L'Hospital's rule, use synthetic long division. Without showing you the details,

(1 - 4n)/(1 - 2n) = [-1/(1 - 2n)] + 2

So we now have a new limit,

lim [-1/(1 - 2n)] + lim(2)

Note that since 2 is a constant, the limit does nothing to it, and we have

lim [-1/(1 - 2n)] + 2

Now, as n goes to infinity, the denominator gets really big, which means the value itself approaches 0. Therefore, our answer is

0 + 2 = 2

Answer 4

1st method: L'Hopital's rule
Differentiate both numerator and denominator because when you substitute infinity into the original expression, you'll get indeterminate form.
So now you have lim of (-4/-2) and when n approaches infinity, the limit is still a constant which is 2!

2nd method: Dividing all terms with highest degree of n by looking at the denominator.
So now you have lim of [(1/n - 4)/(1/n - 2)]. Substituting n as infinity gives you lim of (0-4)/(0-2) which is still a constant which is 2! Because 1/infinity approaches zero.