Evaluating Limits Using L'Hopital's Rule For Lim X→π/2 2cos(7x)sec(5x)

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In the realm of calculus, evaluating limits stands as a fundamental concept, paving the way for understanding continuity, derivatives, and integrals. However, some limits present a unique challenge, particularly those that result in indeterminate forms like 0/0 or ∞/∞. To tackle these perplexing situations, mathematicians turn to a powerful tool known as L'Hôpital's Rule. This article delves into the intricacies of L'Hôpital's Rule and demonstrates its application in evaluating the limit ${\lim _{x \rightarrow \frac{\pi}{2}} 2 \cos (7 x) \sec (5 x)}$.

Understanding L'Hôpital's Rule

L'Hôpital's Rule provides a method for evaluating limits of indeterminate forms. Specifically, if we have a limit of the form ${\lim_{x \to c} \frac{f(x)}{g(x)}}$ where both f(x) and g(x) approach 0 or both approach ±∞ as x approaches c, then L'Hôpital's Rule states that:

${\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}}$

provided the limit on the right-hand side exists. In essence, L'Hôpital's Rule allows us to take the derivative of the numerator and the derivative of the denominator separately and then re-evaluate the limit. This process can be repeated if the new limit is also an indeterminate form.

It's crucial to remember that L'Hôpital's Rule is applicable only to indeterminate forms. Applying it to limits that are not indeterminate will likely lead to incorrect results. Furthermore, the existence of the limit on the right-hand side ${\lim_{x \to c} \frac{f'(x)}{g'(x)}}$ is a condition for the validity of the rule. If this limit does not exist, L'Hôpital's Rule cannot be applied.

Applying L'Hôpital's Rule to ${\lim _{x \rightarrow \frac{\pi}{2}} 2 \cos (7 x) \sec (5 x)}$

Let's now apply L'Hôpital's Rule to evaluate the given limit:

${\lim _{x \rightarrow \frac{\pi}{2}} 2 \cos (7 x) \sec (5 x)}$

To begin, we can rewrite the expression using the identity ${\sec(x) = \frac{1}{\cos(x)}}$:

${\lim _{x \rightarrow \frac{\pi}{2}} 2 \cos (7 x) \sec (5 x) = \lim _{x \rightarrow \frac{\pi}{2}} \frac{2 \cos (7 x)}{\cos (5 x)}}$

As x approaches ${\frac{\pi}{2}}$, we have ${\cos(7 \cdot \frac{\pi}{2}) = \cos(\frac{7\pi}{2}) = 0}$ and ${\cos(5 \cdot \frac{\pi}{2}) = \cos(\frac{5\pi}{2}) = 0}$. Thus, the limit is of the indeterminate form 0/0, making L'Hôpital's Rule applicable.

Next, we differentiate the numerator and the denominator separately:

• Derivative of the numerator: {\(2 \cos (7 x))' = -14 \sin (7 x)}
• Derivative of the denominator: ${(\cos (5 x))' = -5 \sin (5 x)}$

Applying L'Hôpital's Rule, we get:

${\lim _{x \rightarrow \frac{\pi}{2}} \frac{2 \cos (7 x)}{\cos (5 x)} = \lim _{x \rightarrow \frac{\pi}{2}} \frac{-14 \sin (7 x)}{-5 \sin (5 x)} = \lim _{x \rightarrow \frac{\pi}{2}} \frac{14 \sin (7 x)}{5 \sin (5 x)}}$

Now, let's evaluate the limit by substituting x = ${\frac{\pi}{2}}$:

${\frac{14 \sin (7 \cdot \frac{\pi}{2})}{5 \sin (5 \cdot \frac{\pi}{2})} = \frac{14 \sin(\frac{7\pi}{2})}{5 \sin(\frac{5\pi}{2})} = \frac{14(-1)}{5(1)} = -\frac{14}{5}}$

Therefore, the limit is:

${\lim _{x \rightarrow \frac{\pi}{2}} 2 \cos (7 x) \sec (5 x) = -\frac{14}{5}}$

Common Mistakes and How to Avoid Them

When applying L'Hôpital's Rule, several common mistakes can lead to incorrect results. Understanding these pitfalls is crucial for mastering the technique. Here are some key areas to watch out for:

1. Applying the Rule to Non-Indeterminate Forms

The most frequent error is using L'Hôpital's Rule when the limit is not in an indeterminate form (0/0, ∞/∞, 0 ⋅ ∞, ∞ - ∞, 0⁰, 1^∞, or ∞⁰). Always verify that the limit results in an indeterminate form before applying the rule. For instance, if the limit is of the form ${\frac{5}{0}}$, it's not an indeterminate form, and L'Hôpital's Rule is not applicable. Directly evaluating the limit will often yield the correct answer in such cases.

2. Incorrect Differentiation

L'Hôpital's Rule involves differentiating the numerator and the denominator separately. A mistake in differentiation will inevitably lead to a wrong answer. Double-check your derivatives, paying close attention to the chain rule, product rule, and quotient rule when necessary. For example, the derivative of ${\cos(7x)}$ is ${-7\sin(7x)}$, not ${-\sin(7x)}$.

3. Forgetting to Check for Indeterminate Forms After Each Application

After applying L'Hôpital's Rule once, the new limit might still be an indeterminate form. If so, the rule can be applied again. However, it's essential to re-evaluate the limit after each application to ensure it remains an indeterminate form. If the limit is no longer indeterminate, L'Hôpital's Rule should not be applied further.

4. Misunderstanding the Conditions for the Rule

L'Hôpital's Rule has specific conditions that must be met. The functions f(x) and g(x) must be differentiable in an open interval containing c (except possibly at c), and the limit ${\lim_{x \to c} \frac{f'(x)}{g'(x)}}$ must exist. If this limit does not exist, L'Hôpital's Rule cannot be used, and alternative methods for evaluating the limit should be considered.

5. Algebraic Errors

Simplifying the expression after applying L'Hôpital's Rule is crucial. Algebraic errors in simplification can lead to an incorrect final answer. Take the time to carefully simplify the expression, combining like terms and reducing fractions where possible.

6. Circular Reasoning

In some cases, repeated application of L'Hôpital's Rule might lead back to the original indeterminate form, creating a circular pattern. When this happens, L'Hôpital's Rule is not helping to solve the problem, and other techniques, such as algebraic manipulation or trigonometric identities, should be considered.

7. Not Recognizing Alternative Methods

L'Hôpital's Rule is a powerful tool, but it's not always the most efficient method for evaluating limits. Sometimes, algebraic manipulation, trigonometric identities, or other limit laws can provide a quicker and simpler solution. Always consider alternative approaches before resorting to L'Hôpital's Rule.

By being mindful of these common mistakes and actively working to avoid them, you can significantly improve your accuracy and efficiency when using L'Hôpital's Rule.

Alternative Methods for Evaluating Limits

While L'Hôpital's Rule is a valuable tool for evaluating limits, it's not always the most efficient or even the most appropriate method. Several other techniques can be employed, and recognizing when to use them is crucial for mastering limit evaluation. Let's explore some alternative approaches:

1. Algebraic Manipulation

Often, limits can be simplified by algebraic manipulation. This includes techniques like factoring, expanding, combining fractions, and rationalizing the numerator or denominator. For example, consider the limit:

${\lim_{x \to 2} \frac{x^2 - 4}{x - 2}}$

Direct substitution leads to the indeterminate form 0/0. However, factoring the numerator yields:

${\lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}}$

Canceling the common factor (x - 2) simplifies the limit:

${\lim_{x \to 2} (x + 2) = 2 + 2 = 4}$

Algebraic manipulation can often transform an indeterminate form into a determinate one, allowing for direct evaluation.

2. Trigonometric Identities

Trigonometric limits often benefit from the use of trigonometric identities. Identities can help to rewrite the expression in a form that is easier to evaluate. A classic example is the limit:

${\lim_{x \to 0} \frac{\sin(x)}{x}}$

This limit is a fundamental result in calculus and is equal to 1. To evaluate more complex trigonometric limits, identities like the double-angle formulas, Pythagorean identities, and sum-to-product identities can be incredibly useful.

For instance, consider:

${\lim_{x \to 0} \frac{1 - \cos(x)}{x}}$

Multiplying the numerator and denominator by ${1 + \cos(x)}$ and using the identity ${1 - \cos^2(x) = \sin^2(x)}$ can simplify the limit.

3. Special Limit Laws

Certain limit laws provide shortcuts for evaluating specific types of limits. One such law is:

${\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e}$

where e is the base of the natural logarithm. Recognizing this form can quickly solve limits that might otherwise require more complex methods. Similarly, the limit:

${\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e}$

is another important special limit.

4. Squeeze Theorem

The Squeeze Theorem (also known as the Sandwich Theorem) is useful for evaluating limits when the function is bounded between two other functions whose limits are known. If ${g(x) \leq f(x) \leq h(x)}$ for all x near c (except possibly at c), and ${\lim_{x \to c} g(x) = L = \lim_{x \to c} h(x)}$, then ${\lim_{x \to c} f(x) = L}$. A common application is in evaluating limits involving trigonometric functions, particularly when dealing with oscillations.

5. Direct Substitution

Before applying any advanced techniques, it's always wise to try direct substitution. If substituting the value x approaches into the function yields a determinate result, then that is the limit. For example:

${\lim_{x \to 2} (x^2 + 3) = 2^2 + 3 = 7}$

Direct substitution is the simplest method and should be the first approach considered.

6. Series Expansions

For some limits, particularly those involving transcendental functions, using series expansions (such as Taylor or Maclaurin series) can be helpful. Approximating functions with their series expansions near a point can simplify the limit evaluation. For instance, the Maclaurin series for ${\sin(x)}$ is:

${\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots}$

Using this series, the limit ${\lim_{x \to 0} \frac{\sin(x)}{x}}$ can be easily evaluated.

7. Graphical Analysis

Sometimes, graphing the function can provide insights into the limit's behavior. Visualizing the function as x approaches a certain value can help determine if the limit exists and what its value might be. This is particularly useful for functions with discontinuities or oscillatory behavior.

By having a diverse toolkit of limit evaluation techniques, you can approach problems more effectively and efficiently, choosing the method that best suits the given situation.

Conclusion

L'Hôpital's Rule is a powerful tool for evaluating limits of indeterminate forms. However, it's crucial to understand the conditions for its application and to avoid common mistakes. In the case of ${\lim _{x \rightarrow \frac{\pi}{2}} 2 \cos (7 x) \sec (5 x)}$, applying L'Hôpital's Rule allows us to successfully determine the limit as -14/5. Moreover, being familiar with alternative methods for evaluating limits broadens your problem-solving capabilities and enables you to tackle a wider range of limit problems. Mastering these techniques is essential for a solid foundation in calculus and its applications.