Question

Lim x -> a (sqrt(3x) - sqrt(2x + a))/(2(x - a))​

Answer 1

To find the limit of the given expression as x approaches a, we can start by simplifying the expression.

First, let's rationalize the numerator. We can do this by multiplying both the numerator and denominator by the conjugate of the numerator, which is the expression obtained by changing the sign between the two terms.

The conjugate of sqrt(3x) - sqrt(2x + a) is sqrt(3x) + sqrt(2x + a).

By multiplying the numerator and denominator by the conjugate, we get:

((sqrt(3x) - sqrt(2x + a))/(2(x - a))) * ((sqrt(3x) + sqrt(2x + a))/(sqrt(3x) + sqrt(2x + a)))

Expanding the numerator using the difference of squares formula, we get:

(sqrt(3x))^2 - (sqrt(2x + a))^2 = 3x - (2x + a) = x - a

So now our expression becomes:

(x - a)/(2(x - a))

We can see that the numerator and denominator have a common factor of (x - a), so we can cancel them out:

1/2

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

Explanation:

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Answer 2

To find the limit of the given expression, we can apply algebraic manipulations and use the limit properties. Let's break it down step by step:

Step 1: Simplify the expression

Start by simplifying the expression by rationalizing the numerator. Multiply the numerator and denominator by the conjugate of the numerator, which is sqrt(3x) + sqrt(2x + a).

(sqrt(3x) - sqrt(2x + a))/(2(x - a)) * (sqrt(3x) + sqrt(2x + a))/(sqrt(3x) + sqrt(2x + a))

This simplifies the numerator to 3x - (2x + a), which becomes x - a.

Step 2: Cancel out common factors

Now, we can cancel out the common factor of (x - a) in both the numerator and denominator.

(x - a)/(2(x - a))

Step 3: Evaluate the limit

Since the factor (x - a) cancels out, we are left with 1/2. Therefore, the limit of the given expression as x approaches a is 1/2.

lim x -> a (sqrt(3x) - sqrt(2x + a))/(2(x - a)) = 1/2

In summary, the limit of the given expression as x approaches a is 1/2.

~~Alli~~