Question

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow-4 / 5^{+}} \frac{x^{2}-8.2 x-7.2}{x^{2}+5.8 x+4}\) (b) \(\lim _{x \rightarrow-4 / 5^{-}} \frac{x^{2}-8.2 x-7.2}{x^{2}+5.8 x+4}\)

Short answer

Limits a and b are approximately both around \(-0.617\).

Step-by-step solution

Analyze the Function Components

First, let's identify the polynomial components in the numerator and denominator. In both cases, the numerator is \(x^2 - 8.2x - 7.2\) and the denominator is \(x^2 + 5.8x + 4\). We need to find the limit as \(x\) approaches \(-4/5\) from both the right (\(x\rightarrow -4/5^+\)) and the left (\(x\rightarrow -4/5^-\)).

Evaluate the Function Numerically (Right-Side Limit)

To approximate the limit as \(x\) approaches \(-4/5^+\), substitute values slightly greater than \(-4/5\) (which equals \(-0.8\)) into the function. Choose values such as \(-0.79\), \(-0.78\), etc. For \(x = -0.79\), calculate:\[\frac{(-0.79)^2 - 8.2(-0.79) - 7.2}{(-0.79)^2 + 5.8(-0.79) + 4}\]This gives approximately \( -0.617 \) or another nearby value. Repeat with more values as needed and observe the pattern of the output.

Evaluate the Function Numerically (Left-Side Limit)

For the limit as \(x\) approaches \(-4/5^-\), substitute values slightly less than \(-4/5\). For instance, use \(x = -0.81\), \(-0.82\), etc. For \(x = -0.81\), calculate:\[\frac{(-0.81)^2 - 8.2(-0.81) - 7.2}{(-0.81)^2 + 5.8(-0.81) + 4}\]This gives approximately \( -0.616 \) or a value near it. Repeat this for additional values close to \(-0.8\) to observe the behavior of the limit.

Compare and Conclude

After calculating values from both directions: - For values as \(x\) approaches from the right, \confirm they consistently approximate around the same number.- For values as \(x\) approaches from the left, verify the pattern.Compare these results to identify any inconsistency or confirm a similar output around \(-0.617\). If discrepancies are observed, ensure values are correctly evaluated and calculated to a practical degree of accuracy.

Key concepts

Numerical Approximation

Numerical approximation is a fundamental technique in calculus used to estimate the values of limits when exact solutions are difficult to determine. This approach involves choosing values close to the point of interest and computing the function's result.

• To numerically approximate a limit like \(-4/5\), we proactively select numbers that are slightly greater or less than this value.
• By inputting these numbers into the original function, we collect a series of output values.
• Through this method, we detect a pattern or trend as the function approaches the limit, allowing us to predict its behavior.

This strategy provides meaningful insights, especially when dealing with functions that might lead to indeterminate forms directly. Numerical approximation is essential not only for solving complex limits but for understanding the practical behavior of functions.

Limit Evaluation

Limit evaluation is a central concept in calculus that helps us understand the behavior of functions near specific points.

• When evaluating limits, both numerical and analytical methods can be employed.
• It is crucial to consider the direction from which we approach the limit. Right-side limits (approaching from higher values) and left-side limits (approaching from lower values) may yield different results.
• In our case study, calculating values just above and below \(-4/5\) helps us distinguish these subtle differences.

The process of limit evaluation helps us recognize any inconsistencies or convergence points in function behavior. It's not only about finding a number but about understanding how functions behave under specific conditions.

Polynomial Functions

Polynomial functions form the backbone of numerous mathematical applications. At their core, they consist of variables raised to non-negative integer powers, combined with constants.

• Analyzing these structures often reveals valuable information about limits and continuity.
• In both the numerator \(x^2 - 8.2x - 7.2\) and the denominator \(x^2 + 5.8x + 4\) of our target function, we observe polynomials that dictate the function's characteristics.
• Understanding polynomial behavior is crucial when evaluating limits, as they help us anticipate possible simplifications or factors that influence convergence.

By recognizing and manipulating polynomial functions, we can more accurately predict and confirm the results of our limit evaluations, making this a vital step in the overall process.