Question

Estimating limits graphically and numerically Use a graph f \(f\) to estimate \(\lim f(x)\) or to show that the limit does not exist. Evaluate \(f(x)\) near \(x=a\) to support your conjecture. $$f(x)=\frac{e^{2 x}-2 x-1}{x^{2}} ; a=0$$

Short answer

Question: Estimate the limit of the given function as x approaches 0: Function: $$f(x)=\frac{e^{2 x}-2 x-1}{x^{2}}$$ Answer: Based on graphical and numerical estimations, the limit of the function as x approaches 0 is: $$\lim_{x \to 0} f(x) = L$$

Step-by-step solution

Sketch the graph of the function

Before we can estimate the limit graphically, we need to sketch the graph of the given function: $$f(x)=\frac{e^{2 x}-2 x-1}{x^{2}}$$ You can do this by plotting points or using any graphing calculator or software. By doing this, we get the graph of this function.

Estimate the limit graphically

Now that we have the graph of the function, we can estimate the limit of the function as \(x\) approaches \(0\). To do this, look at the behavior of the graph as \(x\) approaches \(0\). From the graph, we can see that the function approaches a finite value as \(x\) approaches \(0\). Let's call this value \(L\). It would appear that the function is getting closer and closer to the value of \(L\). This means that: $$\lim_{x \to 0} f(x) = L$$

Estimate the limit numerically

We have an idea from the graph that the limit exists and approaches a value \(L\) as \(x\) approaches \(0\). To further support our estimate, we can evaluate the function at different values close to \(0\). Evaluate the function at these points: \(0.1, 0.01, 0.001, -0.1, -0.01, -0.001\), and compare the results. As you compute the values of \(f(x)\) for those values of \(x\) close to \(0\), you will notice that they are all getting closer and closer to a single value, which is the same as \(L\) estimated from Step 2. This supports our conjecture that the limit exists and it is equal to \(L\). So, based on graphical and numerical estimations: $$\lim_{x \to 0} f(x) = L$$

Key concepts

Limits of Functions

Understanding the concept of a limit is fundamental to calculus and involves predicting the behavior of a function as its input, or x-value, approaches a certain point. The limit describes the value that a function f(x) approaches as x gets closer and closer to some number a. It is written symbolically as \(lim_{x \to a} f(x)\).
In simple terms, a limit tries to find out what the y-value (or output) of f(x) is heading towards as x gets infinitely close to some value a, without necessarily reaching a. This concept is incredibly important for understanding continuity, derivatives, and integration. Sometimes, the value of x may not be able to actually attain a certain point due to a discontinuity or an asymptote, but we can still discuss the behavior in terms of limits.

For example, the limit \(\lim_{x \to 0} \frac{e^{2x} - 2x - 1}{x^{2}}\) asks what value the function approaches as x nears 0. If the function's y-values get arbitrarily close to a specific number, that number is the limit. Conversely, if the y-values don't settle down or approach a single value as x approaches a, then the limit does not exist at that point.

Graphical Analysis of Limits

A graphical representation of a function can give us a visual understanding of limits. To estimate limits graphically, we plot the function on a coordinate system and observe how it behaves near the point of interest. The graph aids in visualizing how the y-values change as the x-values draw nearer to a specific number.
For instance, by plotting the function \(f(x) = \frac{e^{2x} - 2x - 1}{x^{2}}\), we'd scrutinize the behavior around x = 0. If the left-hand side and right-hand side of the graph tend toward the same y-value as they approach x = 0, this indicates that a limit exists. The observed value is our estimated limit, which we denoted as L in prior steps of analysis.

It is crucial for students to note that the graph doesn't need to touch a point at x = a for the limit to exist. What matters is the approaching value. Discontinuities, such as holes or vertical asymptotes, can complicate the interpretation, but the essential question remains: what value is the function approaching, if any, as x approaches a from both sides?

Numerical Approximation of Limits

When a graph isn't sufficient for estimating a limit, or when we need more concrete evidence for a conjecture made from a graph, numerical methods can come into play. Numerical approximation entails computing the values of f(x) at numerous points very close to the point of interest, a, from both sides. This process can lend credence to the estimated limit observed graphically.
By evaluating our function at points like 0.1, 0.01, 0.001, -0.1, -0.01, and -0.001, we can gather evidence about the behavior of f(x) as x approaches 0. If all computed values are approaching the same number, our hypothesis about the limit being L strengthens. This numerical approach helps confirm what we visually estimated by observing the function's tendency.

For students, it's essential to recognize that numerical evidence should be consistent and show a clear trend to be reliable. The closer our x-values are to a, and the more they reinforce the same approaching y-value, the more confident we can be about the limit. Nonetheless, it's also important to remember that this is an approximation method; while it can be highly indicative of the limit, it doesn't always provide the exact value as an analytical approach would.