Evaluating Limits as x Approaches Infinity: A Step-by-Step Guide

When evaluating the limit of a function as x approaches infinity, especially involving radical expressions, you can use several techniques. This article will guide you through a detailed step-by-step process to evaluate such limits, specifically focusing on the behavior of the difference of two square root functions as x tends to infinity. Let's break down the problem and solution in a clear and understandable manner.

Consider the limit:

lim_{x to infty} (sqrt{x^2-4x} - sqrt{x^2-1})

Step 1: Identifying the Structure

The expression involves the difference of two square root functions. When dealing with such expressions, multiplying and dividing by the radical conjugate can simplify the term and make it easier to evaluate the limit.

Step 2: Applying the Radical Conjugate

Let's start by multiplying the numerator and denominator by the radical conjugate of the numerator:

lim_{x to infty} (sqrt{x^2-4x} - sqrt{x^2-1}) times frac{sqrt{x^2-4x} sqrt{x^2-1}}{sqrt{x^2-4x} sqrt{x^2-1}}

This gives us:

lim_{x to infty} frac{(sqrt{x^2-4x} - sqrt{x^2-1})(sqrt{x^2-4x}   sqrt{x^2-1})}{sqrt{x^2-4x}   sqrt{x^2-1}}

Using the difference of squares formula, the numerator simplifies as follows:

lim_{x to infty} frac{(x^2-4x) - (x^2-1)}{sqrt{x^2-4x}   sqrt{x^2-1}}

This simplifies to:

lim_{x to infty} frac{-4x   1}{sqrt{x^2-4x}   sqrt{x^2-1}}

Step 3: Simplifying the Expression

Now, let's factor out x from the square roots in the denominator:

lim_{x to infty} frac{-4x   1}{x*sqrt{1-frac{4}{x}}   x*sqrt{1-frac{1}{x^2}}}

This further simplifies to:

lim_{x to infty} frac{-4   frac{1}{x}}{sqrt{1-frac{4}{x}}   sqrt{1-frac{1}{x^2}}}

As x approaches infinity, the terms 1/x and 1/x^2 approach 0. Thus, the limit becomes:

lim_{x to infty} frac{-4   0}{sqrt{1-0}   sqrt{1-0}}  frac{-4}{1   1}  -2

However, the correct evaluation should be:

lim_{x to infty} frac{-4   0}{1   1}  -4/2  -2

The correct final answer should be:

-2/2 -1

Final Answer

lim_{x to infty} (sqrt{x^2-4x} - sqrt{x^2-1}) -1

Conclusion

Through the steps above, we have evaluated the limit of the difference of two square root functions as x approaches infinity. By utilizing the radical conjugate and simplifying the expression, the limit can be accurately determined. This method is crucial in solving complex limit problems involving radical expressions. With practice, you can become more proficient in handling such mathematical challenges.