Find the limit.. Calculus?

Question

lim 3x^4 + x - 5 / 6x^4 -2x^2 + 1 as x approaches infinity..

and

lim sqrt(4x^2 + 1) / (3x -1) as x approaches negiative infinity

Answer 1

lim (3x^4 + x - 5)/(6x^4 -2x^2 + 1) = lim (3x^4 -x^2 + 0.5 +
+ x^2 -0.5 + 1)/(6x^4 -2x^2 + 1) =
= lim (3x^4 -x^2 + 0.5)/(6x^4 -2x^2 + 1) +
+ lim (x^2 + 0.5)/(6x^4 -2x^2 + 1) =
lim(3x^4 -x^2 + 0.5))/2(3x^4 -x^2 + 0.5) +
+ lim (x^2 + 0.5)/(6x^4 -2x^2 + 1) = 1/2

lim sqrt(4x^2 + 1) / (3x -1) = lim sqrt(4x^2)/(3x - 1) =
= lim -2x/(3x - 1) = -2/3

Answer 2

I am going to assume that you mean
lim [ 3x^4 + x - 5 ] / [ 6x^4 -2x^2 + 1 ]

As x get very large, this becomes:
lim [ 3x^4 ] / [ 6x^4 ] = lim [ 1/2] = 1/2

For your second problem:
lim sqrt(4x^2 + 1) / (3x -1) -->
lim sqrt(4x^2) / (3x) = lim (2x) / (3x) = 2/3