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Chapter 2: Problem 35

Find the limit, if it exists. $$\lim _{x \rightarrow 0^{+}} \frac{x \sqrt{x}}{x+x^{2}}$$

Short Answer

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Step by step solution

01

Simplify the Given Expression

The given limit is \(\frac{x \sqrt{x}}{x+x^2}\). First, let's simplify the expression. Rewrite the numerator and denominator: \(\frac{x \sqrt{x}}{x(1+x)}\).
02

Cancel Common Factors

Notice that \(x\) is a common factor in both the numerator and the denominator. So, cancel \(x\) from both: \(\frac{\sqrt{x}}{1+x}\).
03

Evaluate the Limit

Now, we evaluate the limit as \(x\) approaches \(0^+\): \(\frac{\sqrt{x}}{1+x}\). Since \(x\) approaches \(0^+\), \(\sqrt{x}\) also approaches \(0\). The denominator \(1+x\) approaches \(1\). Therefore, the limit is \(\frac{0}{1}\).
04

State the Final Answer

As previously calculated, the limit is \(0\). Hence, \(\lim _{x \rightarrow 0^{+}} \frac{x \sqrt{x}}{x+x^{2}} = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
When working with limits, simplifying expressions is a crucial first step. The goal is to make the problem easier to understand and solve. In this exercise, we start with the expression \(\frac{x \sqrt{x}}{x+x^{2}}\). To simplify, we need to rewrite both the numerator and the denominator. Notice that both have a common factor of \(x\). By factoring out \(x\) from the denominator, we get: \(\frac{x \sqrt{x}}{x(1+x)}\).
This simplification helps us see that \(x\) can be canceled from both the numerator and the denominator, leading to a much simpler form: \(\frac{\sqrt{x}}{1+x}\).
Simplifying expressions reduces the complexity and makes evaluating the limit more straightforward.
Limit Evaluation
Evaluating limits involves understanding what happens to a function as the input approaches a particular value. In this case, we are interested in what happens to \(\frac{\sqrt{x}}{1+x}\) as \(x\) approaches \(0^{+}\) (from the positive side).
To find the limit, observe the behavior of both the numerator and the denominator separately:
  • As \(x \rightarrow 0^{+}\), \(\sqrt{x}\) approaches 0.
  • The denominator \(1+x\) approaches 1 because adding a very small number \(x\) to 1 still makes it close to 1.

    • Combining these observations, we have \(\frac{0}{1}\), which results in 0. Therefore, \(\lim_{x \rightarrow 0^{+}} \frac{\sqrt{x}}{1+x} = 0\).
    Basic Calculus
    Basic calculus provides the foundational skills required to solve more complex problems. In the limit problem encountered, we need to apply a few key concepts:
  • Understanding limits and their notation, such as \(\lim_{x \rightarrow 0^{+}}\).
  • Familiarity with simplifying expressions by factoring and canceling common terms.
  • Recognizing the behavior of functions as variables approach specific points.

    • All these principles culminate in the proper evaluation of the limit problem. Calculus teaches us how to break down functions and study their behavior over intervals, which is essential for mastering more advanced mathematical concepts.

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    Most popular questions from this chapter

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