Question

Finding a Limit In Exercises \(41-46,\) write a simpler function that agrees with the given function at all but one point. Then find the limit of the function. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow 0} \frac{x^{4}-5 x^{2}}{x^{2}} $$

Short answer

The limit of the function \( \frac{x^{4}-5 x^{2}}{x^{2}} \) as \( x \) approaches 0 is \( -5 \).

Step-by-step solution

Simplify the function

The first step is to simplify the function. This can be done by factoring out a common factor of \( x^{2} \) from the numerator and then canceling it with the \( x^{2} \) in the denominator. The simplified function is \( x^{2}-5 \).

Evaluate the limit

The next step is to evaluate the limit of the simplified function as \( x \) approaches 0. Substituting \( x = 0 \) into the expression, the result is \( -5 \). Thus, \( \lim _{x \rightarrow 0} \frac{x^{4}-5 x^{2}}{x^{2}}= -5 \).

Confirm the result with a graphing utility

To increase confidence in the solution, a graphing utility can be used to plot the function and visually verify that the function approaches -5 as \( x \) approaches 0. Given the limitations of this solution format, this step can't be shown, but it is recommended to do so.

Key concepts

Limit of a Function

Understanding the limit of a function is fundamental in calculus and showcases what value the function approaches as the input gets closer to a certain point. Essentially, it’s about predicting the behavior of functions at points where they might not be explicitly defined.

Take the given exercise, for example. It requires finding the limit as the variable, in this case, 'x', approaches zero for the function \( \frac{x^4 - 5x^2}{x^2} \). Here, simply plugging in zero for 'x' leads to a problematic expression of \( \frac{0}{0} \), which is undefined. That's where limit analysis comes in - it provides a way to determine the behavior of the function as 'x' gets infinitely close to zero without actually being zero.

To find this limit, we first simplify the function to avoid undefined situations, as was carried out in the exercise. Once simplified, we can then plug in the value of 'x' approaching zero to find the limit, which, in this exercise, is -5. This fundamental methodology enables calculus students to handle functions that aren't straightforward to evaluate at certain points.

Simplifying Expressions

Simplifying expressions is an important skill, making complex calculus problems more manageable. By understanding and identifying common factors, we can alter the form of the function without changing its values at most points, which can lead to a more straightforward analysis of limits.

In our example, \( \frac{x^4 - 5x^2}{x^2} \), simplifying begins by recognizing that \( x^2 \) is a common factor in the numerator. Factoring this out, we get \( x^2(x^2 - 5) \), which, when divided by \( x^2 \) in the denominator, simplifies to \( x^2 - 5 \) after canceling out the common term. This step is critical because it transforms an indeterminate form into one that is easy to evaluate as 'x' approaches zero.

Breaking down expressions into simpler components not only aides in limit evaluation but also helps in other areas of calculus such as differentiation and integration, where working with less complex forms can save time and reduce errors.

Graphing Utility Confirmation

Once simplification and analytical limit-finding are done, it's highly beneficial to confirm results with a graphing tool. This visual aid can confirm what we've calculated by showing the graphical behavior of the function as it approaches the point of interest.

In the exercise, after finding the simplified limit is -5, we further reinforce our confidence by using a graphing utility. While the step-by-step solution does not demonstrate this graphically, it is recommended that students plot the original and simplified functions to see that both approach the same value as 'x' approaches zero.

A good strategy is to look at the plot and see where the function is heading as 'x' gets closer to the point from both left and right sides. If the function approaches the same value from both sides, we have a clear confirmation of our limit. Incorporating this tool is crucial, especially when dealing with more complex or unfamiliar functions, and serves as a strong complement to algebraic techniques in understanding calculus problems.