(i) Lt_(x Rarr 1)/(2))((8x-3)/(2x-1)-(4x^(2)+1)/(4x^(2)-1))

Jasonbradley

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Jun 16, 2024

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Evaluating the Limit: lim_(x→(1/2))((8x-3)/(2x-1)-(4x^(2)+1)/(4x^(2)-1))

This problem asks us to find the limit of a rational function as x approaches 1/2. Let's break down the steps to solve this:

1. Simplifying the Expression

First, we need to simplify the expression within the limit. Notice that the denominator of the second fraction can be factored:

(4x^(2) - 1) = (2x + 1)(2x - 1)

Now we can rewrite the expression as:

(8x - 3)/(2x - 1) - (4x^(2) + 1)/((2x + 1)(2x - 1))

To combine the fractions, we need a common denominator:

[(8x - 3)(2x + 1) - (4x^(2) + 1)] / [(2x + 1)(2x - 1)]

2. Expanding and Simplifying

Let's expand the numerator:

[16x^(2) + 8x - 6x - 3 - 4x^(2) - 1] / [(2x + 1)(2x - 1)]

Combining like terms:

[12x^(2) + 2x - 4] / [(2x + 1)(2x - 1)]

3. Evaluating the Limit

Now we can evaluate the limit as x approaches 1/2. Direct substitution is usually a good first step:

[12(1/2)^(2) + 2(1/2) - 4] / [(2(1/2) + 1)(2(1/2) - 1)]

Simplifying:

[3 + 1 - 4] / [2 * 0] = 0 / 0

We get an indeterminate form (0/0). This means we need to further simplify the expression. Notice that both the numerator and denominator have a common factor of (2x - 1).

Factoring the numerator:

[2(6x^(2) + x - 2)] / [(2x + 1)(2x - 1)]

[2(2x - 1)(3x + 2)] / [(2x + 1)(2x - 1)]

Now we can cancel the (2x - 1) terms:

[2(3x + 2)] / (2x + 1)

4. Final Evaluation

Now we can directly substitute x = 1/2:

[2(3(1/2) + 2)] / (2(1/2) + 1) = [2(5/2)] / 2 = 5/2

Therefore, the limit of the expression as x approaches 1/2 is 5/2.