Problem 91 - Brief Applied Calculus

Question

True or False: \(\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1}=\frac{\lim _{x \rightarrow 1}\left(x^{2}-1\right)}{\lim (x-1)}\)

Step-by-Step Solution

Answer

False.

Step 1: Understand the Limit Expression

The original expression involves the limit of the function \( \frac{x^2-1}{x-1} \) as \( x \to 1 \). This is a rational function, which indicates that direct substitution might result in an undefined form such as \( \frac{0}{0} \).

Step 2: Apply Factoring to Simplify the Function

The expression \( x^2 - 1 \) can be factored as \( (x-1)(x+1) \). This simplifies the function \( \frac{x^2 - 1}{x-1} \) to \( x + 1 \) when \( x eq 1 \).

Step 3: Evaluate the Limit of the Simplified Function

Since the simplified function is \( x + 1 \), we can evaluate the limit as \( x \to 1 \). Therefore, \( \lim_{x \to 1} (x + 1) = 1 + 1 = 2 \).

Step 4: Understand Invalid Limit Division

The expression \( \frac{\lim_{x \to 1} (x^2-1)}{\lim_{x \to 1} (x-1)} \) attempts to separately take the limit of the numerator and the denominator and divide them, which is generally not valid when both approach zero. This results in \( \frac{0}{0} \), an indeterminate form.

Step 5: Correctly State the Comparison

Since the operation of dividing limits individually lead to an indeterminate form, and the correctly computed limit for \( \frac{x^2-1}{x-1} \) is 2, the statement is false.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in calculus that helps simplify complex expressions, making them easier to work with, especially when calculating limits. A polynomial is an expression consisting of variables and coefficients, structured in terms like \(x^2 - 1\). To factor a polynomial means to express it as a product of its simpler components or factors. In our discussed exercise, the polynomial \(x^2 - 1\) can be factored into \((x - 1)(x + 1)\). This process is valuable because it can reveal common factors that allow for simplification in fractions.

By simplifying rational expressions through factoring, we can cancel out terms, often eliminating potential undefined forms like \(\frac{0}{0}\), which can occur when directly substituting values. Factoring not only aids in simplifying expressions but also enhances our understanding of the behavior of functions near specific points, such as approaching a limit.

Indeterminate Forms

An indeterminate form appears in calculus when evaluating a limit results in expressions like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), or other undefined forms. These forms signal that direct substitution of values does not immediately provide a clear solution. Instead, they indicate a need for further manipulation of the expression to evaluate the limit.

In the original exercise, the limit expression \(\frac{x^2-1}{x-1}\) gives rise to \(\frac{0}{0}\) when directly substituting \(x = 1\). This is because both the numerator and the denominator evaluate to zero. To resolve this, we often resort to techniques such as factoring, simplifying, or applying L'Hopital's Rule for differentiation. Through these methods, the indeterminate form can often be transformed into a determinate one with a definable limit result.

Rational Functions

Rational functions are expressions representing the quotient of two polynomials, such as \(\frac{x^2 - 1}{x - 1}\). These functions are central to calculus due to their frequent appearance in problems dealing with limits, derivatives, and integrals. Assessing limits of rational functions helps us understand their behavior as variables approach specific values, influencing the overall curve of their graph.

One common challenge with rational functions is that they can result in undefined values, like division by zero, at specific points. This is where limits play a crucial role, allowing us to discuss the function's behavior just around these problematic points without the expression itself being defined. By analyzing the limit of a rational function, like in our exercise, we strive to learn about the function's behavior as it approaches a particular value, rather than solely focusing on calculations at that point. This ability to study limits broadens our understanding and allows for meaningful analysis beyond mere numeric evaluation.