# 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.