Evaluating Limits Without Using L'H?pital’s Rule: The Case of sin(x) - a / (x - a) cos(x)

Understanding how to evaluate limits is a fundamental skill in calculus. This article explores the evaluation of the limit of the form (lim_{x to a} frac{sin(x) - a}{(x - a) cos(x)}) without relying on L'H?pital's rule. We will also explore another example, (lim_{x to 0} frac{ln(1 - x^2)}{ln(cos x)}), and demonstrate the steps to solve such limits.

Example 1: Evaluating (lim_{x to a} frac{sin(x) - a}{(x - a) cos(x)})

One classical approach to evaluating limits such as (lim_{x to a} frac{sin(x) - a}{x - a cos(x)}) involves rewriting and breaking down the expression into simpler parts. Let's start by using the trigonometric identity and properties of limits to solve this problem.

The limit can be rewritten as:

[lim_{x to a} frac{sin(x) - a}{x - a cos(x)} lim_{x to a} frac{sin(x) - a}{x - a} cdot frac{1}{cos(x)}]

As (x) approaches (a), the expression (frac{sin(x) - a}{x - a}) approaches 1. This is a well-known limit, specifically:

[lim_{x to 0} frac{sin(x)}{x} 1]

By letting (x - a theta), we see that (theta to 0) as (x to a). Therefore, the limit becomes:

[lim_{x to a} frac{sin(x) - a}{x - a} cdot frac{1}{cos(x)} lim_{theta to 0} frac{sin(theta)}{theta} cdot lim_{x to a} frac{1}{cos(x)} 1 cdot frac{1}{cos(a)} sec(a)]

Example 2: Evaluating (lim_{x to 0} frac{ln(1 - x^2)}{ln(cos(x))})

This example involves a more complex limit that can be solved without L'H?pital's rule. We start by rewriting the limit:

[lim_{x to 0} frac{ln(1 - x^2)}{ln(cos(x))} frac{lim_{x to 0} frac{ln(1 - x^2)}{x^2}}{lim_{x to 0} frac{ln(cos(x))}{x^2}}]

We can further simplify the expressions inside the limits:

[lim_{x to 0} frac{ln(1 - x^2)}{x^2} lim_{u to 0} frac{ln(1 - u)}{u}] with (u x^2)

Using the substitution (u x^2), we get:

[lim_{u to 0} frac{ln(1 - u)}{u} lim_{v to infty} frac{lnleft(1 - left(-frac{1}{v}right)right)}{-frac{1}{v}} lim_{v to infty} -frac{lnleft(left(1 frac{1}{v}right)^vright)}{frac{1}{v}} -1]

Similarly, we simplify the denominator:

[lim_{x to 0} frac{ln(cos(x))}{x^2} lim_{x to 0} frac{ln(1 - (1 - cos(x)))}{x^2} lim_{x to 0} frac{ln(1 - (1 - cos(x)))}{1 - cos(x)} cdot frac{1 - cos(x)}{x^2}]

Since (lim_{x to 0} frac{ln(1 - u)}{u} -1) and (lim_{x to 0} frac{1 - cos(x)}{x^2} frac{1}{2}), we get:

[lim_{x to 0} frac{ln(cos(x))}{x^2} -1 cdot frac{1}{2} -frac{1}{2}]

Combining these results, we find:

[lim_{x to 0} frac{ln(1 - x^2)}{ln(cos(x))} frac{-1}{-frac{1}{2}} 2]

Conclusion

Both examples illustrate how to evaluate limits without relying on L'H?pital's rule. Instead, we use algebraic manipulation and known limit properties. By breaking down complex expressions and applying trigonometric identities, we can often simplify the evaluation of such limits.

Key Concepts and Keywords

Limit evaluation: Techniques for finding the limits of functions as variables approach certain values.

L'H?pital's rule: A rule used for evaluating limits of the form 0/0 or (infty/infty).

Trigonometry limits: Special limits involving trigonometric functions that are frequently encountered and proven in calculus.