Problem 43 - Essential Calculus

Question

In Exercises \(43-48,\) find the limit. (Hint: Treat the expression as a fraction whose denominator is 1 , and rationalize the numerator.) Use a graphing utility to verify your result. $$ \lim _{x \rightarrow-\infty}\left(x+\sqrt{x^{2}+3}\right) $$

Step-by-Step Solution

Answer

The limit of the function as x approaches negative infinity is \(0\).

Step 1: Rationalize the numerator

Multiply the expression by its conjugate divided by itself. This will rationalize the numerator. The resulting expression is: \(\lim _{x \rightarrow-\infty}\left((x+\sqrt{x^{2}+3}) * \frac{(x-\sqrt{x^{2}+3})}{(x-\sqrt{x^{2}+3})}\right)\)

Step 2: Simplify the expression rationalized

When multiplied, the conjugate gets rid of the square root in the numerator. The simplified expression will be: \(\lim _{x \rightarrow-\infty}\left(\frac{x^2 - (x^2 + 3)}{x - \sqrt{x^{2}+3}}\right)\)

Step 3: Simplify the expression further

Simplify the numerator, it becomes \(-3\). The resulting expression: \(\lim _{x \rightarrow-\infty}\left(\frac{-3}{x - \sqrt{x^{2}+3}}\right)\)

Step 4: Take the limit

The limit of the function as \(x\) approaches \(-\infty\) is obtained by considering the behaviour of the denominator. As \(x\) approaches \(-\infty\), both terms in the denominator also move towards \(-\infty\), so their subtraction would near zero. Hence the limit is \(0\).

Key Concepts

Rationalizing the Numerator

When working with limits involving square roots, we might encounter indeterminate forms or expressions that are difficult to evaluate. One effective strategy to deal with these scenarios is to rationalize the numerator. This technique involves multiplying the expression by a fraction equivalent to 1, which contains the conjugate of the numerator. By doing this, we eliminate the square root from the numerator, making the limit much easier to handle.

For instance, in the given problem, we are dealing with the limit of an expression as x approaches negative infinity. The expression includes a square root, which complicates matters. To simplify, we multiply the numerator and the denominator by the conjugate pair, which in this example is (x - \(\sqrt{x^2+3}\)). The critical benefit here is that when we multiply conjugate pairs, the square roots cancel out, leaving us with a polynomial in the numerator. This simplification is pivotal for taking the limit at infinity easily.

An improvement advice for learners is to be comfortable with multiplying conjugate pairs and practice recognizing when rationalizing the numerator could simplify the limit problem at hand.

Limits at Infinity

The concept of limits at infinity is essential in understanding the behavior of functions as the input grows without bound. Generally, when finding the limit as x approaches infinity (or negative infinity), the function's values tend to stabilize to a fixed number, which we call the limit. For rational functions, if the degrees of the polynomials in the numerator and denominator are the same, the limit is the ratio of the leading coefficients.

In our exercise problem, after rationalizing the numerator, we get a simplified fraction. As x becomes very large in the negative direction, the terms x and \(\sqrt{x^2+3}\) both grow without bound, but their difference approaches zero. Ultimately, we're left with a situation where a non-zero constant number, in this case, -3, is divided by a value approaching zero. This leads us to the conclusion that the limit is 0. Understanding the behavior of such expressions at infinity helps students predict the behavior of the function without relying solely on graphing utilities.

Conjugate Pairs

When working with square roots in algebra, conjugate pairs come in handy, especially when rationalizing expressions. A conjugate pair consists of two binomials that have the same terms, but opposite operations between them. For example, \(a + \sqrt{b}\) and \(a - \sqrt{b}\) are conjugate pairs.

The magic of conjugate pairs lies in their multiplication. When we multiply conjugate pairs, we get a difference of squares: \( \left( a + \sqrt{b} \right) \left( a - \sqrt{b} \right) = a^2 - b \). This is a powerful tool because it allows us to eliminate square roots, simplifying the expression significantly. In our problem, using the conjugate pair (x - \(\sqrt{x^2+3}\)) to rationalize the numerator is a critical step that simplifies the limit calculation. For students, the key takeaway is recognizing when and how to use conjugate pairs, which often turns a complex limit problem into a much more manageable one.