Understanding and Solving Limits with L'H?pital’s Rule

Introduction to Limits and Indeterminate Forms

In calculus, limits are fundamental in understanding the behavior of functions as they approach certain values. However, sometimes direct substitution can lead to indeterminate forms such as 0/0 or ∞/∞. This is where L'H?pital’s Rule becomes a powerful tool. In this article, we will explore how to apply L'H?pital’s Rule to solve a specific limit problem and understand the process step-by-step.

The Problem: Finding lim x→∞ [√(x21 - x1/x1)]

Let’s consider the limit problem: lim x→∞ [√(x21 - x1/x1)] . When we directly plug infinity into the expression, we get an indeterminate form of type ∞ - ∞. This form does not immediately give us a clear solution, so we need to apply some algebraic manipulation and apply L'H?pital’s Rule.

Applying L'H?pital's Rule

To solve this limit, we can start by simplifying the expression. Let's divide the entire expression by x, both in the numerator and the denominator:

First, let's rewrite the expression: Factor out x from the square root: Divide both the numerator and the denominator by x: Simplify the expression further:

Step-by-Step Solution

Let’s break it down:

Step 1: Rewrite the original expression: lim x→∞ [ √( x21 - x1/x1 ) ] Step 2: Divide by x in the numerator and denominator: lim x→∞ [ ( x21/x - x1/x2 ) / 1 ] Step 3: Simplify the expression: lim x→∞ [ √( x20 - 1/x ) ] Step 4: Apply the square root to each term: lim x→∞ [ x10 √(1 - 1/x21) ] Step 5: Evaluate the limit as x approaches infinity: Step 6: As x approaches infinity, 1/x21 approaches 0: Therefore, the expression simplifies to: lim x→∞ [ x10 √(1 - 0) ] lim x→∞ [ x10 * 1 ] 0

Conclusion

In conclusion, by simplifying the expression and applying the principles of limits and L'H?pital’s Rule, we were able to solve the limit problem and find that the answer is 0. Understanding how to manipulate expressions and apply L'H?pital’s Rule is crucial for dealing with similar problems in calculus.

Further Reading and Resources

To delve deeper into L'H?pital’s Rule and limit solving techniques, consider the following resources:

YouTube tutorials on L'H?pital’s Rule and limits Online calculus textbooks and reference sites Math forums and QA sites for additional practice

If you have any further questions or need additional help, feel free to revisit the resources provided or seek guidance from a tutor or instructor.