Problem 29 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(19-38,\) 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 negative infinity of \(\frac{2 x+1}{\sqrt{x^{2}-x}}\) is \(-2\).

Step 1: Simplify the expression under the limit

Express \(\sqrt{x^{2}-x}\) as \(x\sqrt{1-\frac{1}{x}}\) since \(x^2\) can be taken common from \(\sqrt{x^{2}-x}\). Therefore, the limit will become \(\lim _{x \rightarrow-\infty} \frac{2 x+1}{x\sqrt{1-\frac{1}{x}}}\). Supposedly, we multiply and divide the expression by \(-1\) because \(x\) is tending to \(-\infty\), so \(x\) is negative. Therefore, the limit will be further simplified as \(\lim _{x \rightarrow-\infty} -\frac{2 x+1}{-x\sqrt{1-\frac{1}{x}}}\)

Step 2: Cancel out similar terms

Observe that there is a \(x\) in both the numerator and the denominator. We can simplify by canceling out the \(x\) term, getting \(\lim _{x \rightarrow-\infty} -\frac{2 +\frac{1}{x}}{\sqrt{1-\frac{1}{x}}}\)

Step 3: Take the limit

Now, if we take limit as \(x \rightarrow-\infty\), \(\frac{1}{x} \rightarrow 0\). So the expression becomes \(-\frac{2 +0}{\sqrt{1-0}} = -\frac{2}{1} = -2\)

Key Concepts

Finding Limits

In calculus, finding limits is a fundamental concept that deals with predicting the value that a function approaches as the input (or the variable) approaches a certain value. This concept is crucial because it helps to understand the behavior of functions that are not easily evaluated at specific points, especially when dealing with continuous changes.

The process of finding a limit involves looking at the values of a function as it gets closer to the point of interest. In the given exercise, we are looking at how \( \frac{2x+1}{\sqrt{x^2-x}} \) behaves as \( x \) approaches negative infinity (\( -\infty \)). In practical step-by-step solutions, we systematically simplify the function and use limit laws to find the value it approaches. The exercise improvement advice would suggest to always pay attention to how forms like \( \frac{1}{x} \) become negligible when \( x \) approaches \( \pm\infty \), which simplifies the calculation of the limit.

Limit Properties

The limit properties are a set of rules that make the computation of limits more systematic and manageable. These properties include the sum, difference, product, and quotient of limits, as well as the power and root of limits. They ensure that if we know the limit of two functions separately, we can easily find the limit of a combination of those functions.

For example, one of the properties tells us that the limit of a sum is the sum of the limits. This means that if we have a function like \( f(x) + g(x) \), and we know \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \), then \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \). This property helps simplify complex functions into simpler parts where limits can be determined individually then combined. This is particularly evident in the step-by-step solution where we break down the original expression into parts we can manage.

Infinite Limits

When dealing with infinite limits, we are looking at what happens to a function as the variable approaches infinity or negative infinity. Unlike regular finite limits, the values that functions approach can be unbounded, indicating that they grow without limit.

In the provided exercise, we are dealing with a limit where \( x \) approaches \( -\infty \), which is a classic case of an infinite limit. Understanding the behaviors such as how polynomials dominate radicals or logarithms as \( x \) goes to infinity, or how division by an increasingly large \( x \) tends to zero, is key—both of which are crucial for interpreting the solution to our problem. Especially when the variable goes to negative infinity, we must be mindful of the sign of the terms to correctly determine the direction of the function's end behavior.

Rationalizing Expressions

The technique of rationalizing expressions is especially important when working with limits that involve radicals. Rationalization helps to eliminate the radical in the denominator of a fraction, making it easier to simplify and find the limit.

In the context of the given problem, rationalizing expressions allows us to deal with the square root in a way that we can apply limit laws more effectively. The step-by-step solution demonstrated this by redefining \( \sqrt{x^2-x} \) as \( x\sqrt{1-\frac{1}{x}} \). This transformation is a form of rationalizing an expression, where we factor out the common term from under the radical to simplify the process of finding the limit. Commonly in exercises of this type, multiplying by a conjugate is another method of rationalization, which can be applied depending on the nature of the radical expression involved in the limit.