Understanding and Solving Complex Algebraic Expressions

The question at hand requires us to solve a complex algebraic expression involving square roots and simplifications. We will break down the problem into manageable steps and explain each process clearly.

Simplifying the Expression

Consider the expression:

sqrt[x - 3^2 3] - sqrt[x - 1^2 4]

First, let's simplify the constants within the square roots:

sqrt[x - 9 3] - sqrt[x - 1 4]

Which simplifies to:

sqrt[x - 6] - sqrt[x 3]

Calculating the Limits

To find the limits, we will consider the expression as x approaches positive and negative infinity.

For x approaching positive infinity:

We can use the alternative approach to simplify the expression:

L lim (x - 3^2 3)^(1/2) - (x - 1^2 4)^(1/2)

As x gets very large, the terms x - 6 and x 3 dominate the square roots:

L lim (x - 6)^(1/2) - (x 3)^(1/2)

L lim x - 3 - (x 3)^(1/2)

L x - 3 - (x 3)^(1/2)

Subtracting the limits:

L lim x - 3 - x - 1 -2

For x approaching negative infinity:

Similarly, for large negative x:

L 3 - (-x) - (-x) 1

L 3 x x 1

L 4 2x

Which approaches 2 as x tends to -∞:

L 2

Alternative Approach: Rationalizing the Numerator

Another effective method is to rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator:

(sqrt[x - 3^2 3] - sqrt[x - 1^2 4]) * (sqrt[x - 3^2 3] sqrt[x - 1^2 4]) / (sqrt[x - 3^2 3] sqrt[x - 1^2 4])

This will eliminate the square roots in the numerator:

(x - 9 3 - (x - 1 4)) / (sqrt[x - 3^2 3] sqrt[x - 1^2 4])

Simplifying further:

(x - 6 - (x 3)) / (sqrt[x - 3^2 3] sqrt[x - 1^2 4])

(-9) / (sqrt[x - 3^2 3] sqrt[x - 1^2 4])

As x approaches positive or negative infinity, the denominator grows indefinitely, leaving the limit as:

L -2 for positive infinity and L 2 for negative infinity.

Shortcut for Competitive Exams

To solve such problems quickly in competitive exams, remember:

sqrt(x^2) |x|

This simplifies the expressions involved in the limit calculation. For large positive x, the dominant terms are:

lim (x - 3) - (x - 1) -2

For large negative x, the dominant terms are:

lim (3 - (-x)) - ((-x) - 1) 2

Thus, the problem can be quickly solved using the properties of absolute values and dominant terms.