Question

Let \(a\) and \(b\) be positive real numbers. Evaluate \(\lim _{x \rightarrow \infty}(a x-\sqrt{a^{2} x^{2}-b x})\) in terms of \(a\) and \(b\)

Short answer

Answer: The limit is 0, as is simplified in the step-by-step solution: $$\lim _{x \rightarrow \infty}(a x-\sqrt{a^{2} x^{2}-b x}) = 0$$.

Step-by-step solution

Write the given expression

We are given the expression: $$\lim _{x \rightarrow \infty}(a x-\sqrt{a^{2} x^{2}-b x})$$

Factor out \(x\) from the expression inside the square root

By factoring out \(x\) from the expression inside the square root, we get: $$\lim _{x \rightarrow \infty}(a x-\sqrt{x^2(a^{2}-\frac{b}{x})})$$

Factor out \(x^2\) from the expression inside the square root

By factoring out \(x^2\) from the expression inside the square root, we get: $$\lim _{x \rightarrow \infty}(a x-\sqrt{x^2( a^2-\frac{b}{x} }\,))$$

Evaluate the square root

By evaluating the square root, we have: $$\lim _{x \rightarrow \infty}(a x-x\sqrt{a^{2}-\frac{b}{x}})$$

Factor out \(x\) from the entire expression

We can now factor out \(x\) from the entire expression: $$\lim _{x \rightarrow \infty}x(a-\sqrt{a^{2}-\frac{b}{x}})$$

Divide both terms inside the limit by \(x\)

To simplify the limit, we can divide both terms inside the limit by \(x\). This will cancel out the \(x\) we factored out. $$\lim _{x \rightarrow \infty}x \frac{ a-\sqrt{a^{2}-\frac{b}{x}} }{ x }$$

Evaluate the limit as x approaches infinity

As \(x\) approaches infinity, the term \(\frac{b}{x}\) approaches 0. This results in the limit of the expression as: $$\lim _{x \rightarrow \infty}x(a-\sqrt{a^{2}-0 })$$ The expression inside the limit becomes: $$x(a-\sqrt{a^{2}})$$ Since \(\sqrt{a^2} = a\), we obtain: $$x(a-a)$$ Thus, the limit is simply 0 since all terms within the expression will be cancelled out: $$\lim _{x \rightarrow \infty}(a x-\sqrt{a^{2} x^{2}-b x}) = 0$$

Key concepts

Real Numbers

Real numbers are the set of numbers that include all the rational and irrational numbers. This means they consist of whole numbers, fractions, and decimals, both terminating and non-terminating. Their value can be expressed along the continuum of the number line.
A few key points to remember about real numbers are:

• They include numbers like 0, positive and negative integers (e.g., -3, 0, 1), fractional numbers (e.g., 1/2, 3.76), and irrational numbers (e.g., π, √2).
• Real numbers can be used to represent a distance along a line.
• When considered in the context of calculus, real numbers provide necessary values that can be approached but not surpassed, especially in limits.

In our problem, real numbers are represented by the variables 'a' and 'b', which add flexibility to understanding the limit of a given expression.

Infinity

Infinity, in a mathematical sense, is a concept that describes something endless or limitless. In calculus, it often appears in limits, integrals, and series, providing a way to talk about quantities that grow larger and larger without bound.

• When you say a variable such as 'x' approaches infinity, it means that as it increases, it goes beyond any finite number.
• Considering the behavior at infinity helps understand the asymptotic nature of functions and how they behave at the edges of their domain.
• In our exercise, infinity is crucial, as the limit expression is evaluated as x approaches infinity to see how the function behaves when x becomes very large.

Remember, infinity itself isn't a number but a way of expressing unboundedness in the context it's used.

Square Roots

Square roots involve finding a number which, when multiplied by itself, gives the original number. Symbolically, if you have a number x, its square root is represented as \( \sqrt{x} \).

• When working with square roots in calculus, especially in limits, it's important to consider simplification to make evaluation easier.
• Factors inside square roots are often manipulated to help simplify expressions, an example of which is used in our exercise when \( \sqrt{a^{2} x^{2}-b x} \) is considered.
• It's important to recognize properties of square roots like \( \sqrt{x^2} = |x| \) to simplify correctly in any algebraic context.

Understanding how to handle square roots is essential in calculus as they frequently appear when evaluating, differentiating, or integrating functions.

Factorization

Factorization in mathematics refers to breaking down a composite number or expression into a product of simpler numbers or terms.

• In calculus, factorization is a useful technique to simplify expressions, making it possible to evaluate limits, derivatives, or integrals more easily.
• In our problem, expressions such as \( x^2(a^{2}-\frac{b}{x}) \) show the importance of factorization for simplifying the limit. By factoring, we transitioned complex expressions into simpler ones.
• This process helps cancel out terms, making the remaining parts of the expression easier to manage and interpret, which is critical when dealing with limits as in our original exercise.

Factorization is a key algebraic skill that provides clarity and simplicity amid complex mathematical expressions.