Introduction to Limits and Derivatives

In calculus, limits are fundamental to understanding the behavior of functions as they approach certain values. Similarly, derivatives describe the rate of change of functions at different points. This article will explore a specific limit problem and demonstrate how to solve it through algebraic manipulation and the concept of derivatives.

Problem and Initial Approach

The problem we'll focus on is the following limit:

( lim_{x to 3} frac{sqrt{x 1}-2}{x-3} )

Initially, substituting (x 3) into the function results in an indeterminate form:

( frac{sqrt{3 1}-2}{3-3} frac{2-2}{0} frac{0}{0} )

This indeterminate form, (frac{0}{0}), suggests that we need to manipulate the expression further to find the limit.

Algebraic Manipulation and Conjugate Method

To simplify the function, we can multiply the numerator and the denominator by the conjugate of the numerator:

( frac{sqrt{x 1}-2}{x-3} cdot frac{sqrt{x 1} 2}{sqrt{x 1} 2} frac{(sqrt{x 1})^2-2^2}{(x-3)(sqrt{x 1} 2)} frac{x-3}{(x-3)(sqrt{x 1} 2)} )

Since (x - 3) appears in both the numerator and the denominator, we can cancel them out, provided (x eq 3):

( lim_{x to 3} frac{1}{sqrt{x 1} 2} frac{1}{sqrt{3 1} 2} frac{1}{2 2} frac{1}{4} )

Evaluating the Limit Using Derivatives

Another approach involves recognizing that the given limit is related to the definition of the derivative. Let's define (x-3 h). Then the limit becomes:

( lim_{h to 0} frac{sqrt{4h 1}-2}{h} lim_{h to 0} frac{sqrt{4h 1}-sqrt{4}}{h} )

This form is familiar as it matches the definition of the derivative of (f(x) sqrt{x 1}) at (x 3). The derivative of (sqrt{x 1}) is:

( f'(x) frac{1}{2sqrt{x 1}} )

Thus, evaluating the derivative at (x 3):

( f'(3) frac{1}{2sqrt{3 1}} frac{1}{2 cdot 2} frac{1}{4} )

Conclusion and Further Exploration

In summary, we found that the limit:

( lim_{x to 3} frac{sqrt{x 1}-2}{x-3} frac{1}{4} )

This exploration not only reinforces the concept of limits and derivatives but also demonstrates the power of algebraic manipulation and the definition of derivatives. Understanding these concepts is crucial for further studies in calculus and other advanced mathematical topics.