Understanding the Existence of Limits: A Detailed Analysis of (lim_{x to 1} x^2 sqrt{x} - 1)

In the realm of calculus and mathematical analysis, the concept of limits is fundamental. One common question that often arises is whether a specific limit exists. This article delves into the details of determining the existence of the limit (lim_{x to 1} x^2 sqrt{x} - 1).

Introduction to Limits in Calculus

Before we proceed with the analysis, it is important to understand the basics of limits. A limit of a function describes the behavior of the function as the input (variable) approaches a certain value. For a limit to exist, the function must approach a specific, finite value as the variable approaches the point of interest.

Evaluating the Limit (lim_{x to 1} x^2 sqrt{x} - 1)

Let's evaluate the limit (lim_{x to 1} x^2 sqrt{x} - 1) step-by-step to determine if it exists and, if so, what its value is.

Step 1: Direct Substitution

One common method to evaluate limits is by direct substitution. If substituting the value directly into the function does not yield an indeterminate form (such as division by zero or an undefined expression), the limit is the value obtained.

Let's substitute (x 1) into the function (x^2 sqrt{x} - 1):

[1^2 sqrt{1} - 1 1 times 1 - 1 1 - 1 0]

Since the substitution yields a finite value, we can conclude that the limit exists and is equal to 0.

Step 2: Verifying the Continuity

Another way to check if the limit exists is to verify the continuity of the function at the point (x 1). A function is continuous at a point if the limit of the function as (x) approaches that point is equal to the value of the function at that point.

For the function (f(x) x^2 sqrt{x} - 1), let's check the continuity at (x 1):

[f(1) 1^2 sqrt{1} - 1 0]

Since the limit as (x) approaches 1 is 0, which is also the value of the function at (x 1), the function is continuous at (x 1).

Alternative Scenarios and Limit Existence

Scenario 1: Function Not Defined for (x

It is possible that the function is not defined for values of (x

Scenario 2: Limit as (x) Approaches 1 from the Right ((x to 1^ ))

Another scenario to consider is the limit as (x) approaches 1 from the right, denoted as (x to 1^ ). This means that (x) is greater than 1 but arbitrarily close to 1. Since the function is defined and continuous in this vicinity, the limit will be the same as the direct substitution:

[ lim_{x to 1^ } x^2 sqrt{x} - 1 0 ]

Conclusion

Based on the analysis, we can conclude that the limit (lim_{x to 1} x^2 sqrt{x} - 1) exists and is equal to 0. The function is continuous at (x 1) and the direct substitution method confirms this.

Therefore, the final answer is:

[ boxed{lim_{x to 1} x^2 sqrt{x} - 1 0} ]