Problem 25 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { x \rightarrow 0 } \frac { \sin x } { 2 x ^ { 2 } - x }$$

Step-by-Step Solution

Answer

The limit as \(x\) approaches 0 for the function \(\frac { \sin x } { 2 x ^ { 2 } - x }\) does not exist both graphically and algebraically.

Step 1: Graphical Determination

Graph the given function \(\frac { \sin x } { 2 x ^ { 2 } - x }\) and observe the behavior of the function as the x-value approaches zero. It's worth noting that for \(x=0\), the function is undefined. However, we can look at the function around \(x=0\) instead. If the function approaches a particular value when x is very close to zero, that value is the limit.

Step 2: Algebraic Determination

For algebraically determining the limit, we can't directly substitute \(x=0\) in the function because it would lead to division by zero. We can, however, rewrite the function to \(\frac { \sin x } { x(2x - 1) }\) and apply the limit theorem.

Step 3: Apply Limit Theorem

Now apply the limit theorem to each part of the function. The limit of \(sinx\) as \(x\) approaches zero is zero. To determine the limit of the denominator as \(x\) approaches zero, directly substitute \(x=0\) to get 0. This gives us a form of \(0/0\) which is indeterminate. However, as \(x\) approaches zero from the right side, it's more rigorous to take the limit as \(x\) approaches zero from the right side, or \( \lim _ { x \rightarrow 0^{+}} \frac { x(2x - 1) }\). This limit happens to be 0. As the function changes behavior around \(x=0\), the limit as \(x\) approaches \(0\) does not exist as it is not the same from the left and right leading to an undefined behavior in the graph.

Key Concepts

Graphical Analysis

Graphical analysis is a strong visual approach to understanding the behavior of functions as they approach a specific value. In this case, we examine the function \(\frac{\sin x}{2x^2 - x}\) to see how it behaves as \(x\) gets closer to zero. Visualizing the graph can be quite helpful because it allows for an immediate sense of the function's limit. By plotting the function, observe what happens near the point of interest, which is \(x = 0\). Here, it's essential to move closely but not precisely to the point of undefined behavior.The graph shows that as \(x\) nears zero from either side, the function shoots off to positive or negative infinity or exhibits different directional approaches, indicating the absence of a single limit. This supports our discovery of its undefined nature at \(x = 0\), reinforcing the idea that graphical analysis provides a good initial insight before engaging in algebraic calculations.

Algebraic Limits

Algebraic limits involve utilizing algebraic techniques to find the limit of a function without relying on a graph. Here, for the function \(\frac{\sin x}{2x^2 - x}\), substituting \(x = 0\) results in \(\frac{0}{0}\), an undefined form, clearly indicating a need for further steps.To proceed, simplify the expression by factoring the denominator as \(x(2x - 1)\). This step reveals more about the function's structure. The goal is to evaluate the limit by understanding each component independently and look for simplifications or approaches that resolve the undefined form. It may require L'Hôpital's Rule or series expansion, depending on complexity. In this simplified condition, directly looking into related limits such as \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) may offer insight for clearer results.

Indeterminate Forms

The term 'indeterminate forms' refers to expressions where standard arithmetic doesn't hold, often appearing as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). In our function \(\frac{\sin x}{2x^2 - x}\), evaluating directly at \(x = 0\) yields \(\frac{0}{0}\). To handle such cases, advanced methods such as L'Hôpital's Rule or algebraic manipulation can be applied to find meaningful results.The function complicates the result by presenting different behaviors as \(x\) approaches zero from either side. This particular behavior—an indeterminate like \(0/0\)—often requires considering the behavior by approaching \(x\) from both positive and negative directions separately. This way, while solving analytically, you can confirm a limit exists or establish that it doesn’t based on consistent direct results from either side of the zero point. Recognizing indeterminate forms in these circumstances is crucial for tackling limits analytically and understanding the subtleties involved in functions approaching certain values.