For which of these functions f(x) does limit as x approaches negative infinity f(x)=2
A. (x-2)/(3x-5)
B. 2x/sqrt(x-2)
C. (2x^2-6x+1)/(1+x^2)
D. (2x-1)/(x^2+1)
E. None of these
as x gets large, the c reduces to
2x^2-6x+1/x^2 which reduces to
2+0+0
To find the function that satisfies the condition, we evaluate the limits of each function as x approaches negative infinity.
A. (x-2)/(3x-5):
To find the limit, divide every term in the numerator and denominator by x:
lim(x->-∞) [(x/x) - (2/x)] / [(3x/x) - (5/x)]
As x approaches negative infinity, the terms with x in the denominator go to zero:
lim(x->-∞) (1 - 0) / (3 - 0) = 1/3
The limit is not equal to 2. Therefore, option A is not correct.
B. 2x/sqrt(x-2):
As x approaches negative infinity, the denominator becomes ∞ and the numerator also becomes ∞, resulting in an indeterminate form.
To evaluate this limit, we can multiply both the numerator and denominator by the conjugate of the denominator, sqrt(x-2):
lim(x->-∞) [2x * sqrt(x-2)] / [sqrt(x-2) * sqrt(x-2)]
The denominator simplifies to x-2, and both the numerator and denominator approach ∞ as x approaches -∞:
lim(x->-∞) [2x * sqrt(x-2)] / (x - 2)
To solve this limit, we can divide every term by x:
lim(x->-∞) [2 * sqrt(x-2)] / (1 - 2/x)
As x approaches -∞, the second term in the denominator goes to 0:
lim(x->-∞) [2 * sqrt(x-2)] / 1 = ∞
The limit is not equal to 2. Therefore, option B is not correct.
C. (2x^2-6x+1)/(1+x^2):
Divide every term in the numerator and denominator by x^2:
lim(x->-∞) [(2x^2/x^2) - (6x/x^2) + (1/x^2)] / [(1/x^2) + (x^2/x^2)]
As x approaches negative infinity, the terms with x in the denominator go to zero:
lim(x->-∞) (2 - 0 + 0) / (0 + 1)
The limit is equal to 2. Therefore, option C is correct.
D. (2x-1)/(x^2+1):
Divide every term in the numerator and denominator by x^2:
lim(x->-∞) [(2x/x^2) - (1/x^2)] / [(x^2/x^2) + (1/x^2)]
As x approaches negative infinity, the terms with x in the denominator go to zero:
lim(x->-∞) (0 - 0) / (1 + 0) = 0
The limit is not equal to 2. Therefore, option D is not correct.
Based on the evaluations of the limits:
The function f(x) that satisfies the given condition is: C. (2x^2-6x+1)/(1+x^2)
To find the limit as x approaches negative infinity for each function, we can analyze the highest power of x in the numerator and denominator. Let's go through each option:
A. (x-2)/(3x-5)
The highest power of x is 1 in both the numerator and the denominator. As x approaches negative infinity, the fraction will approach 2/3, not 2. Therefore, option A is not correct.
B. 2x/sqrt(x-2)
Again, the highest power of x is 1 in the numerator and 1/2 in the denominator. As x approaches negative infinity, the fraction will approach positive infinity, not 2. Therefore, option B is not correct.
C. (2x^2-6x+1)/(1+x^2)
In this case, the highest power of x is 2 in both the numerator and the denominator. When x approaches negative infinity, the fraction will approach 2/1 = 2. Therefore, option C is correct.
D. (2x-1)/(x^2+1)
The highest power of x is 1 in the numerator and 2 in the denominator. When x approaches negative infinity, the fraction will approach 0/1 = 0, not 2. Therefore, option D is not correct.
Considering the above analysis, the correct answer is option C, and none of the other options satisfy the given condition.