Problem 23 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{2 x+1}{\sqrt{x^{2}-x}} $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches \(-\infty\) of \(\frac{2x+1}{\sqrt{x^{2}-x}}\) is \(-2\).

Step 1: Rationalize the denominator

To begin with, multiply both the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate is \(\sqrt{x^{2}+x}\). Make sure to distribute in the numerator and simplify the denominator using difference of squares.

Step 2: Simplify the function

After step 1, the function becomes \((2x+1)\sqrt{x^{2}+x}/(x^{2}-x^2) = (2x+1)\sqrt{x^{2}+x}\). Simplify it further by canceling out terms to leave \(2x+1\) in the numerator and the square root term in the denominator.

Step 3: Evaluate the limit

Next, divide every term by \(x\), simplify and apply the limit as \(x\) approaches \(-\infty\) to get \(\lim _{x \rightarrow-\infty} 2+\frac{1}{\sqrt{1-\frac{1}{x}}}\). Since \(-\infty\) in the denominator equals to zero, we can replace \(\frac{1}{x}\) with zero, therefore, \(\lim_{x\rightarrow-\infty}\frac{2x+1}{\sqrt{x^{2}-x}}=-2\).

Key Concepts

Calculus and Limits

Calculus is a branch of mathematics that focuses on rates of change (differential calculus) and accumulation of quantities (integral calculus). Within calculus, understanding limits is foundational. A limit describes the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits can provide crucial information about a function's behavior, and are often used to determine the continuity, derivatives, and integrals of functions.

When we consider the limit of a function as the variable approaches negative infinity, as in the given exercise, we explore the behavior of the function over a large scale. Calculating such limits involves algebraic manipulations and understanding how large values of variables affect the function's output. The end goal is to determine if the function approaches a specific value, or if it diverges, growing without bound. In this exercise, the final conclusion is that the function approaches the limit -2 as the variable approaches negative infinity.

Rationalizing the Denominator

Rationalizing the denominator is an algebraic technique used to eliminate radicals (such as square roots) from the denominator of a fraction. In calculus problems involving limits, particularly those with an infinite limits context, this technique can be quite useful.

When simplifying the expression  frac{x^2}{x^2-x}, you multiply by the conjugate of the denominator, leading to a more manageable, non-radical form. This tactic allows for clearer observation in how each term behaves as the variable approaches large values, negative or positive. The process also involves simplifying the result, often revealing the core behavior of the function asymptotically. It's essential that students get comfortable rationalizing denominators, as it frequently simplifies calculus expressions, making finding limits far more straightforward.

Infinite Limits

Infinite limits explore the concept of a function's value increasing or decreasing without bound as the independent variable approaches a certain point or infinity. Unlike finite limits where the function approaches a particular number, an infinite limit suggests that the function goes off to infinity or negative infinity.

When a function's value becomes unbounded as we look at inputs farther and farther in the positive or negative direction, we can say that it has an infinite limit in that direction. For instance, in the provided exercise, the limit  frac{2x+1}{ ஹயh ஹயh   â ஹய- ஹய- ஹயn   ⪠ஹய⪠ஹயh ஹய- ஹய-n h   ஹh  ஹh ஹ â      ื to negative infinite confirms that behavior. Mastering infinite limits is not just a necessity for calculus students, but also a skill that can be applied to understand real-world phenomena that show exponential growth or decay—like certain financial models or population studies.