Evaluating Limits A Step By Step Guide

In the realm of calculus, limits play a foundational role, serving as the bedrock upon which concepts like continuity, derivatives, and integrals are built. Understanding how to evaluate limits is therefore crucial for anyone venturing into the world of advanced mathematics. In this article, we will delve into the process of evaluating a specific limit problem, exploring various techniques and strategies along the way. Our focus will be on the function ${ \frac{\sqrt{1+x} - 3}{x - 8} }$ as x approaches 8. This particular limit presents an interesting challenge because direct substitution leads to an indeterminate form, necessitating the use of algebraic manipulation to arrive at a solution. This exploration will not only provide a step-by-step solution but also illuminate the underlying principles and methods applicable to a broader range of limit problems. Mastering these techniques is essential for anyone seeking a deeper understanding of calculus and its applications in various fields.

Our primary task is to evaluate the limit of the function ${ \frac{\sqrt{1+x} - 3}{x - 8} }$ as x approaches 8. This can be mathematically expressed as:

${ \lim_{x \to 8} \frac{\sqrt{1+x} - 3}{x - 8} }$

At first glance, one might be tempted to employ the straightforward approach of direct substitution. However, substituting x = 8 directly into the function yields:

${ \frac{\sqrt{1+8} - 3}{8 - 8} = \frac{\sqrt{9} - 3}{0} = \frac{3 - 3}{0} = \frac{0}{0} }$

The expression ${ \frac{0}{0} }$ is an indeterminate form. This means that direct substitution fails to provide a meaningful answer, and we must resort to alternative methods to determine the limit. Indeterminate forms signal the need for algebraic manipulation or other techniques to simplify the expression and reveal the true behavior of the function as x approaches the given value. This is a common scenario in limit problems, and recognizing indeterminate forms is the first crucial step in choosing the appropriate solution strategy. In the following sections, we will explore a powerful technique known as rationalization, which will help us circumvent this indeterminate form and successfully evaluate the limit.

Since direct substitution resulted in an indeterminate form, we need a different approach to evaluate the limit. One common and effective technique for dealing with expressions involving square roots is rationalization. In this specific problem, we will rationalize the numerator of the function. Rationalizing the numerator involves multiplying both the numerator and the denominator by the conjugate of the numerator. The conjugate of ${ \sqrt{1+x} - 3 }$ is ${ \sqrt{1+x} + 3 }$. This seemingly simple step is a clever algebraic trick that helps eliminate the square root in the numerator, thereby simplifying the expression and making it amenable to limit evaluation.

The rationale behind this technique lies in the difference of squares factorization: ${ (a - b)(a + b) = a^2 - b^2 }$. By multiplying the numerator by its conjugate, we transform the expression into a form where the square root is eliminated, and we can potentially cancel out terms that were causing the indeterminate form. This method is particularly useful when dealing with limits involving square roots, as it often unveils the underlying structure of the function and allows us to proceed with the limit evaluation. In the next section, we will apply this technique to our specific problem and see how it simplifies the expression.

To rationalize the numerator, we multiply both the numerator and denominator of the function by the conjugate of the numerator, which is ${ \sqrt{1+x} + 3 }$:

${ \lim_{x \to 8} \frac{\sqrt{1+x} - 3}{x - 8} \times \frac{\sqrt{1+x} + 3}{\sqrt{1+x} + 3} }$

This multiplication yields:

${ \lim_{x \to 8} \frac{(\sqrt{1+x} - 3)(\sqrt{1+x} + 3)}{(x - 8)(\sqrt{1+x} + 3)} }$

Now, we apply the difference of squares formula, ${ (a - b)(a + b) = a^2 - b^2 }$, to the numerator:

${ \lim_{x \to 8} \frac{(1+x) - 3^2}{(x - 8)(\sqrt{1+x} + 3)} }$

Simplifying the numerator, we get:

${ \lim_{x \to 8} \frac{1 + x - 9}{(x - 8)(\sqrt{1+x} + 3)} }$

Further simplification gives us:

${ \lim_{x \to 8} \frac{x - 8}{(x - 8)(\sqrt{1+x} + 3)} }$

Now, we can cancel out the ${ (x - 8) }$ terms in the numerator and denominator, which were causing the indeterminate form:

${ \lim_{x \to 8} \frac{1}{\sqrt{1+x} + 3} }$

With the problematic term eliminated, we can now proceed to evaluate the limit by direct substitution. This simplification is a crucial step in the process, as it transforms the original indeterminate expression into one that can be easily evaluated. In the next section, we will perform the direct substitution and arrive at the final answer.

After successfully rationalizing the numerator and simplifying the expression, we are now in a position to evaluate the limit using direct substitution. Our simplified expression is:

${ \lim_{x \to 8} \frac{1}{\sqrt{1+x} + 3} }$

Substituting x = 8 into the expression, we get:

${ \frac{1}{\sqrt{1+8} + 3} = \frac{1}{\sqrt{9} + 3} = \frac{1}{3 + 3} = \frac{1}{6} }$

Therefore, the limit of the function as x approaches 8 is ${ \frac{1}{6} }$. This result provides a clear and definitive answer to the problem, demonstrating the effectiveness of the rationalization technique in resolving indeterminate forms. The process of direct substitution, once the expression has been simplified, is a straightforward way to obtain the final value of the limit. This step highlights the importance of algebraic manipulation in preparing the expression for evaluation.

In this comprehensive exploration, we successfully evaluated the limit of the function ${ \frac{\sqrt{1+x} - 3}{x - 8} }$ as x approaches 8. The initial attempt at direct substitution led to the indeterminate form ${ \frac{0}{0} }$, which necessitated the application of a more sophisticated technique. We employed the method of rationalizing the numerator, a powerful algebraic tool that allowed us to simplify the expression by eliminating the square root in the numerator. This involved multiplying both the numerator and denominator by the conjugate of the numerator, which transformed the expression into a form where we could cancel out the problematic term ${ (x - 8) }$.

After simplification, we were able to evaluate the limit by direct substitution, arriving at the final answer of ${ \frac{1}{6} }$. This exercise underscores the importance of algebraic manipulation in evaluating limits, particularly when dealing with indeterminate forms. Recognizing these forms and applying appropriate techniques, such as rationalization, are crucial skills in calculus. The process of evaluating limits is not just about finding a numerical answer; it's about understanding the behavior of functions as they approach certain values. The techniques and strategies discussed in this article provide a foundation for tackling a wide range of limit problems and developing a deeper appreciation for the concepts of calculus.

Limits, rationalization, indeterminate form, direct substitution, conjugate, algebraic manipulation, calculus, function, square root, numerator, denominator, difference of squares, evaluation, problem-solving, mathematics