Evaluating Limits Without L'H?pital's Rule: A Detailed Approach

In this article, we explore how to evaluate the following limit without using L'H?pital's Rule:

(lim_{x to 1} 1 - x tanleft(frac{pi x}{2}right))

Behavior of Components as (x to 1)

First, let's analyze the behavior of each component as (x) approaches 1.

Tangent Function Behavior

As (x) approaches 1, (frac{pi x}{2}) approaches (frac{pi}{2}). The tangent function has a vertical asymptote at (frac{pi}{2}), causing (tanleft(frac{pi x}{2}right) to infty) as (x to 1^-).

Linear Component Behavior

The linear term (1 - x) approaches 0 as (x) approaches 1.

Combining these behaviors, the limit takes the form (0 cdot infty), which is indeterminate. To resolve this, we can rewrite the limit as a quotient:

(lim_{x to 1} 1 - x tanleft(frac{pi x}{2}right) lim_{x to 1} frac{1 - x}{cotleft(frac{pi x}{2}right)})

Rewriting the Cotangent Term

Since (cotleft(frac{pi x}{2}right) frac{1}{tanleft(frac{pi x}{2}right)}), and as (tanleft(frac{pi x}{2}right) to infty) as (x to 1), we have (cotleft(frac{pi x}{2}right) to 0). Thus, the new limit form is (frac{0}{0}), which is also indeterminate.

Using Taylor Series Expansion

To evaluate the limit, we use the Taylor series expansion of (tanleft(frac{pi x}{2}right)) around (frac{pi}{2}):

(tanleft(frac{pi x}{2}right) approx frac{1}{frac{pi}{2} - frac{pi x}{2}} frac{2}{pi (1 - x)} text{ as } x to 1)

Substituting this into our limit:

(lim_{x to 1} 1 - x cdot frac{2}{pi (1 - x)} approx lim_{x to 1} frac{2 (1 - x)}{pi (1 - x)} lim_{x to 1} frac{2}{pi} frac{2}{pi})

Alternative Method Using Substitution

Alternatively, we can set (x 1 - y) and consider the limit:

(lim_{y to 0} frac{y sin(1 - pi y / 2)}{cos(1 - pi y / 2)} lim_{y to 0} frac{y sin(1 - pi y / 2)}{sin(y pi / 2)} frac{2}{pi})

Here, we express (tan(frac{pi x}{2})) as (frac{sin(frac{pi x}{2})}{cos(frac{pi x}{2})}). By rewriting the denominator using trigonometric identities, we simplify the limit expression.

Conclusion

In conclusion, without using L'H?pital's Rule, we have determined that the limit evaluates to (boxed{frac{2}{pi}}).

By understanding the behavior of the trigonometric functions and using series expansions, we can evaluate indeterminate forms and find the exact limit value.