Evaluating Limits And Simplifying Expressions A Comprehensive Guide

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In the realm of calculus, evaluating limits stands as a fundamental concept, serving as the bedrock for understanding continuity, derivatives, and integrals. Limits describe the behavior of a function as its input approaches a specific value. When we delve into the expression $\lim_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2}$, we embark on a journey to decipher the function's inclination as x gravitates towards 7. This particular limit presents an interesting challenge due to its indeterminate form, which necessitates the application of advanced techniques to arrive at a definitive solution.

The Indeterminate Form and Rationalization

Direct substitution of x = 7 into the expression yields an indeterminate form of 0/0, a clear indication that further manipulation is required. This predicament often arises when dealing with limits involving radicals. A common strategy to tackle such situations is rationalization, a technique that eliminates radicals from the numerator or denominator. In this case, we employ a clever approach involving the multiplication of the numerator and denominator by conjugate expressions. The conjugate expressions are carefully crafted to eliminate the radicals through the difference of squares or cubes factorization patterns. This process transforms the original expression into a more manageable form, allowing us to circumvent the indeterminate form.

To begin, let's identify the problematic radical terms: $\sqrt{x+2}$, $\sqrt[3]{x+20}$, and $\sqrt[4]{x+9}$. To rationalize these terms, we'll multiply both the numerator and denominator by a series of conjugate expressions. This meticulous process aims to eliminate the radicals one by one, gradually simplifying the expression. The key lies in recognizing the appropriate conjugate for each radical term and applying the corresponding algebraic manipulation.

Step-by-Step Rationalization Process

  1. Rationalizing the Numerator: We start by rationalizing the numerator, focusing on the terms $\sqrt{x+2} - \sqrt[3]{x+20}$. To eliminate the square root, we multiply by the conjugate $\sqrt{x+2} + \sqrt[3]{x+20}$. This gives us $(x+2) - (x+20)^{2/3}$ in the numerator. Then we will eliminate the cubic root by multiplying by $(x+2)^{2/3} + (x+20)^{1/3}(x+2) + (x+20)$.
  2. Rationalizing the Denominator: Next, we shift our attention to the denominator, specifically the term $\sqrt[4]{x+9} - 2$. To rationalize this, we multiply by the conjugate $\sqrt[4]{x+9} + 2$, resulting in $\sqrt{x+9} - 4$. We repeat this process by multiplying by $\sqrt{x+9} + 4$ to eliminate the square root.

This meticulous rationalization process, while algebraically intensive, paves the way for simplifying the limit expression and ultimately evaluating it.

L'Hôpital's Rule: A Powerful Tool

An alternative approach to evaluating limits of indeterminate forms is L'Hôpital's Rule. This powerful rule states that if the limit of a quotient of two functions results in an indeterminate form (0/0 or ∞/∞), then the limit of the quotient of their derivatives may exist and be equal to the original limit. Mathematically, if $\lim_{x \to c} f(x) / g(x)$ is of the form 0/0 or ∞/∞, then $\lim_{x \to c} f(x) / g(x) = \lim_{x \to c} f'(x) / g'(x)$, provided the limit on the right-hand side exists.

Applying L'Hôpital's Rule to the Limit

In our case, the limit $\lim_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2}$ is of the indeterminate form 0/0. Therefore, we can apply L'Hôpital's Rule. This involves taking the derivative of the numerator and the denominator separately.

  1. Derivative of the Numerator: The derivative of $\sqrt{x+2} - \sqrt[3]{x+20}$ is $\frac{1}{2\sqrt{x+2}} - \frac{1}{3(x+20)^{2/3}}$.
  2. Derivative of the Denominator: The derivative of $\sqrt[4]{x+9} - 2$ is $\frac{1}{4(x+9)^{3/4}}$.

Now, we consider the limit of the quotient of these derivatives: $\lim_{x \to 7} \frac{\frac{1}{2\sqrt{x+2}} - \frac{1}{3(x+20)^{2/3}}}{\frac{1}{4(x+9)^{3/4}}}$. Substituting x = 7 into this expression yields a numerical value, which represents the limit of the original expression.

Evaluating the Derivatives at x = 7

Plugging in x = 7 into the derivatives, we get:

  • Numerator derivative at x = 7: $\frac{1}{2\sqrt{7+2}} - \frac{1}{3(7+20)^{2/3}} = \frac{1}{6} - \frac{1}{3(9)} = \frac{1}{6} - \frac{1}{27}$
  • Denominator derivative at x = 7: $\frac{1}{4(7+9)^{3/4}} = \frac{1}{4(16)^{3/4}} = \frac{1}{4(8)} = \frac{1}{32}$

Therefore, the limit becomes: $\lim_{x \to 7} \frac{\frac{1}{6} - \frac{1}{27}}{\frac{1}{32}} = \frac{\frac{7}{54}}{\frac{1}{32}} = \frac{7}{54} * 32 = \frac{112}{27}$

Thus, by applying L'Hôpital's Rule, we find that the limit of the expression as x approaches 7 is 112/27.

Simplifying expressions is a crucial skill in mathematics, enabling us to represent complex equations in a more concise and manageable form. Consider the expression $\frac{5}{\sqrt[5]{1} - \frac{3}{1}}$. This expression involves radicals and fractions, and simplifying it involves evaluating the radicals and performing the arithmetic operations.

Step-by-Step Simplification

The key to simplifying this expression lies in understanding the properties of radicals and fractions. The fifth root of 1, denoted as $\sqrt[5]{1}$, is simply 1, as any root of 1 is always 1. This simplification immediately transforms the expression into a more manageable form.

  1. Evaluating the Radical: $\sqrt[5]{1} = 1$
  2. Substituting the Value: Replacing $\sqrt[5]{1}$ with 1, the expression becomes $\frac{5}{1 - 3}$.
  3. Performing Subtraction: The denominator simplifies to 1 - 3 = -2$.
  4. Final Simplification: The expression now becomes $\frac{5}{-2}$, which can be written as $-\frac{5}{2}$.

Therefore, the simplified form of the expression $\frac{5}{\sqrt[5]{1} - \frac{3}{1}}$ is -5/2. This process highlights the importance of understanding fundamental mathematical principles in simplifying complex expressions.

Conclusion

In conclusion, evaluating limits and simplifying expressions are essential skills in mathematics. Evaluating the limit $\lim_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2}$ requires careful application of rationalization techniques or L'Hôpital's Rule, ultimately leading to the solution of 112/27. On the other hand, simplifying the expression $\frac{5}{\sqrt[5]{1} - \frac{3}{1}}$ involves understanding the properties of radicals and fractions, resulting in the simplified form of -5/2. Mastering these techniques is crucial for success in calculus and beyond.