Problem 98 - Calculus of a Single Variable

Question

HOW DO YOU SEE IT? Would you use the dividing out technique or the rationalizing technique to find the limit of the function? Explain your reasoning. (a) \(\lim _{x \rightarrow-2} \frac{x^{2}+x-2}{x+2} \quad\) (b) \(\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}\)

Step-by-Step Solution

Answer

For problem (a), the dividing out technique would be used because the function is a rational function and it can be easily factored and simplified by dividing out common factors. As for problem (b), the rationalizing technique is most convenient because the function contains a square root. Thus, rationalizing the numerator (multiplying by its conjugate over itself) allows the function to be simplified and the limit to be found.

Step 1: Problem A: Dividing Out Technique

Looking at the structure of the function, it is noticed that the function is rational and can be factored. The dividing out technique is best suited for this. First, simplify the function by factoring and cancelling out any common factors that aren't zero. Starting with the function: \(\lim _{x \rightarrow-2} \frac{x^{2}+x-2}{x+2}\), it can be factored to yield \(\lim _{x \rightarrow-2} \frac{(x-1)(x+2)}{x+2}\). It can be seen that \(x+2\) is a common factor and can be cancelled out, giving: \(\lim _{x \rightarrow-2} (x-1)\). Plug in the limit to get the result.

Step 2: Problem B: Rationalizing Technique

In this problem, the function contains a square root operation in the numerator. The rationalizing technique is most appropriate here. First rationalize the expression by multiplying by the conjugate of the numerator over itself. Starting with \(\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}\), multiply by \(\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}\) to get: \(\lim _{x \rightarrow 0} \frac{x}{x(\sqrt{x+4}+2)}\). The denominator becomes \(x+\sqrt{x+4}+2\). Plugging in the limit value of zero for x, yields the final result.

Key Concepts

Dividing Out Technique

When evaluating limits, particularly those involving rational functions, the dividing out technique is a valuable tool. This method is applied when a function has a numerator and denominator where common factors can be canceled out. The primary goal is to simplify the expression by removing these common factors.

Let's illustrate this with an example: consider the limit \[\lim _{x \rightarrow-2} \frac{x^{2}+x-2}{x+2}.\]By factoring the polynomial in the numerator, \(x^2 + x - 2\) becomes \((x-1)(x+2)\). This reveals that \(x+2\) is a common factor in both the numerator and the denominator, allowing us to cancel it out. Simplification leads to:\[\lim _{x \rightarrow-2} (x-1).\]After canceling, substitute \(x = -2\) into the simplified expression to find the limit.

In problems where direct substitution leads to an indeterminate form like \(\frac{0}{0}\), the dividing out technique simplifies the expression, thus removing the indeterminate form and providing a clearer path to finding the limit.

Rationalizing Technique

The rationalizing technique is crucial when dealing with limits involving square roots. Often, direct substitution results in an indeterminate form, necessitating an algebraic manipulation to simplify the expression successfully.

Consider the following limit:\[\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}.\]Detect the potentially challenging square root in the numerator. To alleviate this, multiply the numerator and denominator by the conjugate of the numerator:\[\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}.\]This operation helps simplify the radical expression by applying the difference of squares formula, resulting in a real number in the numerator and potentially reducing or eliminating the \(x\) in the denominator.

Through rationalizing, we achieve a form that no longer poses the problem of indeterminate form upon substituting the limit value of \(x\). The expression becomes easier to handle and evaluates to a concrete value.

Factoring Polynomials

Factoring polynomials is a valuable skill, especially in the realm of calculus and limits. It transforms complex polynomial expressions into simpler pieces, making them more manageable.

Polynomials like \(x^2 + x - 2\) can often be factored into products of simpler binomials. Here, it breaks down into \((x-1)(x+2)\). This transparency is critical when using techniques like the dividing out technique, where canceling common factors is necessary.

• Identify if the polynomial can be factored – first check for common monomials, then assess patterns like difference of squares or special products like quadratic trinomials.
• Once factored, observe common factors that may be present with other parts of the expression, like the denominator in a fraction.

Factoring is often the stepping stone required to transform an indeterminate form into one that can be recalculated into a definite limit. It's a fundamental technique in simplifying rational expressions.

Simplifying Rational Expressions

In calculus, simplifying rational expressions is often pivotal for finding limits, especially when dealing with complex or indeterminate forms.

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying means reducing them to their most elementary form, typically by factoring and canceling out common factors. Doing this clarifies the expression and allows for a more straightforward evaluation of limits.

• Factor polynomials involved in the rational expression to reveal and cancel out common terms that can simplify the fraction.
• Ensure there are no remaining factors that could lead to an indeterminate form, especially when substituting the limit value.

This simplification directly influences the accuracy and ease of limit evaluation, often smoothing a path around the pitfalls of direct substitution, which might otherwise result in undefined or indeterminate forms.