Problem 59 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated limit, if it exists. \(\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}\) Hint: Multiply by \(\frac{\sqrt{x}+1}{\sqrt{x}+1}\)

Step-by-Step Solution

Answer

The limit exists and it is equal to \(\frac{1}{2}\).

Step 1: Write down the given function and the hint

Given function is: \( \lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1} \) Hint: Multiply by \( \frac{\sqrt{x}+1}{\sqrt{x}+1} \)

Step 2: Utilize the hint and multiply the given function

Now, we will multiply the function by the hint expression: \( \lim_{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1} \times \frac{\sqrt{x}+1}{\sqrt{x}+1} \)

Step 3: Perform the multiplication

Multiply the numerators and denominators: \( \lim_{x \rightarrow 1} \frac{(\sqrt{x}-1)(\sqrt{x}+1)}{(x-1)(\sqrt{x}+1)} \) Using the difference of squares formula: \( (a-b)(a+b) = a^2 - b^2 \), we can simplify the numerator: \( \lim_{x \rightarrow 1} \frac{x-1}{(x-1)(\sqrt{x}+1)} \)

Step 4: Cancel common terms in the numerator and the denominator

We see that there is a common term of (x-1) in both the numerator and the denominator; we can cancel this term out: \( \lim_{x \rightarrow 1} \frac{1}{\sqrt{x}+1} \)

Step 5: Evaluate the limit

Finally, we can evaluate the limit by substituting x with the limit value (1) into the simplified expression: \( \frac{1}{\sqrt{1}+1} = \frac{1}{1+1} = \frac{1}{2} \) The limit exists, and it is equal to \( \frac{1}{2} \).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and accumulation. It is divided chiefly into two areas: differential calculus, which studies the rate at which quantities change, and integral calculus, which studies the accumulation of quantities. Limits, which we encounter in the calculation of derivatives, are a fundamental part of calculus. Understanding limits allows us to handle situations where values approach a certain point but never quite reach it, often yielding insights into the behavior of functions near those points.

Limit of a Function

In calculus, the limit of a function is a fundamental concept that describes the behavior of the function as the input approaches a particular value. The notation
\( \lim_{x \rightarrow a} f(x) \) represents the limit of the function \(f(x)\) as \(x\) approaches the value \(a\). If the function \(f(x)\) approaches a specific value \(L\) as \(x\) gets arbitrarily close to \(a\), then the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\). A well-defined limit is essential for developing the concepts of continuity, derivatives, and integrals in calculus.

Difference of Squares

The difference of squares is an algebraic pattern that represents a specific way to factor certain kinds of polynomials. Specifically,
\(a^2 - b^2 = (a+b)(a-b)\).
This formula allows us to simplify expressions where a term is squared and subtracted from another squared term. In the context of limits, the difference of squares can be particularly useful for simplifying complex rational functions, enabling further simplification and often helping to resolve indeterminate forms such as \(\frac{0}{0}\).

Simplifying Expressions

Simplifying expressions is a critical skill in both algebra and calculus, involving the manipulation of equations or expressions to make them easier to understand or work with. The process often includes expanding multiplication, factoring, canceling common terms, and reducing fractions to their simplest form. When working with limits in calculus, simplifying an expression is a common strategy for finding the value of the limit, especially when direct substitution results in an indeterminate form. Simplification can reveal the underlying behavior of the function as it approaches the limiting value.