Problem 38 - Calculus of a Single Variable

Question

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow(-1 / 2)^{+}} \frac{6 x^{2}+x-1}{4 x^{2}-4 x-3} $$

Step-by-Step Solution

Answer

The step-by-step solution will provide the final answer. The result will be the value of the function when x equals -1/2, unless it's confirmed in the first step that the denominator equals zero when x equals -1/2.

Step 1: Check for denominator equal to zero

Find values of x for the denominator \(4x^2 - 4x - 3 = 0\). This will help us know at which values the function is undefined.

Step 2: Substitute x if denominator is not zero

If Step 1 confirms that \(x = -1/2\) is not a solution to the denominator equation, directly substitute \(-1/2\) into the function \(\frac{6x^2 + x - 1}{4x^2 - 4x - 3}\)

Step 3: Evaluate the limit

Compute the value of the function at \(x = -1/2\) and that will be the one-sided limit.

Key Concepts

Limit of a Function

The concept of a limit is foundational in calculus and helps in understanding the behavior of functions as they approach a specific point. A limit attempts to find the value that a function approaches as the input (or 'x' value) approaches some number. In mathematical terms, when we write \( \lim_{x \rightarrow c} f(x) \), we want to know what value 'f(x)' is getting closer to as 'x' nears the number 'c'.

One key aspect of limits is understanding that they are not necessarily the value of the function at 'c' but rather the value it approaches. One-sided limits, like the one in our exercise, look at this approach either from the left-an \( x \rightarrow c^{-} \) or from the right—an \( x \rightarrow c^{+} \) of the point 'c'. This distinction is crucial when the function has a different behavior on each side of 'c' or when 'c' is a point of discontinuity, meaning the function isn't defined at that exact value of 'x'.

Understanding one-sided limits can be essential in determining properties such as continuity and can help us handle cases where functions have jumps, holes, or vertical asymptotes, which is often the case with rational functions.

Calculus

Calculus is a branch of mathematics that deals with change and motion. It comprises two major areas: differential calculus, which is concerned with the concept of a derivative, and integral calculus, focused on the concept of an integral. Calculus provides tools, like the limit, to rigorously define instantaneous rates of change and to compute areas under curves.

The process of finding a limit, as demonstrated in our one-sided limit exercise, is part of differential calculus. Differential calculus delves into how we can approximate the slope of a curve at a point, which leads to understanding rates of change such as velocity and acceleration. The principles of calculus are widely applied across various fields, including physics, engineering, economics, and biology. Its ability to break down complex systems into understandable parts by looking at infinitesimal changes makes it a powerful tool for solving real-world problems.

Rational Functions

Rational functions are ratios of two polynomials. They are expressed in the form \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. The behavior of a rational function is heavily influenced by its denominator, \( q(x) \), since it determines the function's domain. A notable feature of rational functions is that they can have discontinuities—which are points where the function is not defined.

In the context of our exercise, the denominator has been set to zero to find these points of discontinuity. Because a function cannot have a real value where the denominator is zero, these points are often where we might find vertical asymptotes or holes in the graph of the function.

Rational functions can model a wide range of phenomena in the real world where ratios are relevant, such as rates of work, odds in probability, and concentration in chemistry. Grasping the behavior of these functions around their discontinuities is often pivotal, as demonstrated by the importance of computing one-sided limits.