Understanding the Limit of (sqrt{1-x} - 1) / x as x Approaches 0: Explained with Two Methods

The problem at hand is to find the limit of ((sqrt{1-x} - 1) / x) as (x) approaches 0. This question is important in calculus and real analysis, and it can be solved using two methods: direct substitution and L'H?pital's rule.

Method 1: Direct Substitution and L'H?pital's Rule

The expression ((sqrt{1-x} - 1) / x) results in an indeterminate form 0/0 when (x) approaches 0. Hence, we can apply L'H?pital's rule. This rule states that if the limit of the ratio of two functions as (x) approaches a point is an indeterminate form, then the limit of their derivatives will yield the same result, if the derivatives exist and the limit of the derivatives exists.

Using L'H?pital's rule, we differentiate the numerator and the denominator separately:

[lim_{x to 0} frac{sqrt{1-x} - 1}{x} lim_{x to 0} frac{frac{d}{dx} (sqrt{1-x} - 1)}{frac{d}{dx} x} lim_{x to 0} frac{frac{1}{2sqrt{1-x}} cdot (-1)}{1} frac{-1}{2sqrt{1-x}} bigg|_{x to 0} frac{1}{2}]

Thus, the limit is ( frac{1}{2} ).

Method 2: Simplifying with Algebraic Manipulation

Another approach involves algebraic manipulation to simplify the expression. By multiplying the numerator and the denominator by ((sqrt{1-x} 1)), we get:

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

Substituting (x 0) into the simplified expression:

[lim_{x to 0} frac{-1}{sqrt{1-x} 1} frac{-1}{sqrt{1-0} 1} frac{-1}{2} cdot (-1) frac{1}{2}]

Hence, the limit is still ( frac{1}{2} ).

Conclusion

Both methods confirm that the limit of ((sqrt{1-x} - 1) / x) as (x) approaches 0 is ( frac{1}{2} ).

Understanding the methods and applying them step-by-step is crucial in solving mathematical limits. This example demonstrates the utility of both direct substitution and L'H?pital's rule in handling indeterminate forms.

Related Keywords

limit as x approaches 0 L'H?pital's rule mathematical limit

Further Reading

For more information, you may want to explore topics such as calculus, real analysis, and problem solving techniques. Understanding these concepts will help you tackle more complex problems in mathematics.