If the limit as x approaches infinity (6x^2/200-4x-kx^2)=1/2, then k=
A. 3
B. -3
C. 12
D. -12
E. -3
lim [ 6x^2 / (200-4x-kx^2) ] as x->infinity = 1/2
Notice that the numerator and denominator has the same highest degree of x. Therefore, we can use the L'hopital's rule. To do this, we divide both numerator and denominator by x^2:
lim [ (6x^2 / x^2) / (200-4x-kx^2)/x^2 ] as x->infinity = 1/2
lim [ 6 / (200/x^2 - 4/x - k) ] as x->infinity = 1/2
Substituting x = infinity,
= 6 / ( 0 - 0 - k ) = 1/2
= 6 / (-k) = 1/2
Therefore,
k = -12
hope this helps~ `u`
*Correction*
I'm sorry, the one I did is not called L'hopital's rule. That is just another technique of finding limits (dividing numerator and denominator by the variable with highest degree).
L'hopital's rule is different, for it uses separate differentiation of numerator and denominator when the answer is you're getting is indeterminate, if x in the limit is substituted. Actually, we can also use it here (assuming you've already covered derivatives in class):
derivative of 6x^2 = 12x
derivative of 200 - 4x - kx^2 = -4 - 2kx
If we substitute x = infinity, we'll still get an answer of indeterminate. Thus we use the rule again:
derivative of 12x = 12
derivative of -4 - 2kx = -2k
Rewriting,
lim 12 / -2k as x->infinity = 1/2
Substituting x = infinity,
= 12 / -2k = 1/2
= 6 / -k = 1/2
Thus, k = -12
I hope that's clear now.
Sorry about that, I think I need to get some sleep now. ^^;
Well, let's first simplify the expression. We have (6x^2)/(200-4x-kx^2). As x approaches infinity, the highest power of x in the denominator becomes dominant, so we can ignore all other terms. Hence, the expression becomes (kx^2)/(kx^2) = 1.
Now, since the denominator is the same as the numerator, we can cancel out the x^2 terms. This leaves us with k/k = 1. Therefore, k must equal 1.
However, none of the answer choices match 1, so my best recommendation is to consult your mathematical notes or ask a human math tutor for assistance. Good luck!
To find the value of k, we need to determine the coefficient of x^2 in the given expression as x approaches infinity.
First, simplify the expression:
6x^2 / (200 - 4x - kx^2)
Divide every term in the expression by x^2:
6 / (200/x^2 - 4/x - k)
As x approaches infinity, both 4/x and 200/x^2 approach 0. So the expression simplifies to:
6 / (-k)
We are given that the limit of this expression is 1/2. So we can set up the following equation:
1/2 = 6 / (-k)
To solve for k, we can cross-multiply and simplify:
2 * 6 = -k
12 = -k
Therefore, k = -12. Hence, the answer is (D) -12.
To find the value of k that satisfies the given limit:
First, let's simplify the expression:
lim(x→∞) (6x^2 / (200 - 4x - kx^2))
To determine the value of k, we need to find the highest power of x in the numerator and denominator. In this case, both the numerator and denominator have the highest power of x as 2.
Now, divide both the numerator and denominator by the highest power of x, which is x^2:
lim(x→∞) (6x^2 / (200 - 4x - kx^2))
= lim(x→∞) (6 / ((200 / x^2) - (4x / x^2) - k))
As x approaches infinity, the values 4x / x^2 and 200 / x^2 become smaller and smaller, tending towards 0. Therefore, we can simplify further:
lim(x→∞) (6 / ((200 / x^2) - (4x / x^2) - k))
= lim(x→∞) (6 / (0 - 0 - k))
= lim(x→∞) (6 / (-k))
Since the limit is given as 1/2, we have:
1/2 = 6 / (-k)
To solve for k, we need to isolate it. Multiply both sides of the equation by -k:
-1 * 1/2 = -1 * (6 / (-k))
-1/2 = 6 / k
Cross-multiply:
-1 * k = 2 * 6
-k = 12
Divide both sides by -1:
k = -12
Therefore, the value of k that satisfies the given limit is -12, which corresponds to option D.