Question

Evaluate the given limit. $$ \lim _{x \rightarrow \pi} \frac{x^{2}+3 x+5}{5 x^{2}-2 x-3} $$

Short answer

The limit is approximately 0.6061.

Step-by-step solution

Identify the form of the limit

First, substitute \(x = \pi\) into the expression to identify if it leads to an indeterminate form. The numerator becomes \(\pi^2 + 3\pi + 5\) and the denominator becomes \(5\pi^2 - 2\pi - 3\). Both are non-zero, so there is no indeterminate form.

Substitute and Evaluate

Since direct substitution does not lead to an indeterminate form, calculate the actual values by substituting \(\pi\) into the numerator and denominator separately. For the numerator: \(\pi^2 + 3\pi + 5\). For the denominator: \(5\pi^2 - 2\pi - 3\).

Simplify and Calculate the Limit

Perform the calculations: The numerator simplifies to approximately \(\pi^2 + 3\pi + 5 \approx 9.8696 + 9.4248 + 5 = 24.2944\). The denominator becomes \(5\pi^2 - 2\pi - 3 \approx 49.3480 - 6.2832 - 3 = 40.0648\). Thus, the limit is approximately \(\frac{24.2944}{40.0648}\).

Finalize the Result

Divide the numerically simplified values: \(\frac{24.2944}{40.0648} \approx 0.6061\). Therefore, the limit evaluates to approximately \(0.6061\).

Key concepts

Indeterminate Forms

In calculus, an indeterminate form is a mathematical expression that does not lead to a clear limit and requires further analysis to resolve. Some common indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), \( 1^\infty \), and \( 0^0 \). These forms arise when substituting values directly into a limit results in an undefined situation. In such cases, additional techniques—like algebraic manipulation or L'Hôpital's rule—are employed to evaluate the limit.

In the given problem, substituting \( x = \pi \) into the function does not produce an indeterminate form, as both the numerator and denominator are non-zero. This simplifies the process significantly, allowing us to directly compute the limit without additional manipulation. Thus, whenever you face a limit problem, begin by checking for indeterminate forms to understand how to proceed.

Direct Substitution

Direct substitution is a straightforward method used in limit evaluation. It involves placing the value that \( x \) approaches directly into the function. If this substitution does not result in an indeterminate form, such as \( \frac{0}{0} \), the limit can be evaluated directly using this method.

For the exercise, substituting \( x = \pi \) into the function \( \frac{x^{2}+3x+5}{5x^{2}-2x-3} \) resulted in a non-zero numerator and denominator. Therefore, direct substitution is valid here. This makes the process quicker and easier, as no additional calculations or considerations are needed aside from arithmetic operations to find the limit. Always try direct substitution first—it often simplifies the evaluation significantly.

Numerical Approximation

When evaluating limits, especially those involving complex numbers or transcendental functions, numerical approximation is a valuable tool. It provides an approximate value of the limit when exact calculations are tedious or complex. Here, we evaluate each part of the fraction by substituting \( \pi \) or its decimal approximation \( 3.1416 \).

For the given limit, the numerator approximately evaluates to \( 24.2944 \) and the denominator to \( 40.0648 \). These values are obtained by using an approximate value for \( \pi \) in the polynomial equations. While this method doesn't give an exact result, it's often sufficient for practical purposes and gives a good estimate of the limit's value. Numerical approximation proves particularly useful when calculator or computational help is available.

Limit Evaluation

The final step in limit evaluation involves simplifying the expression you get after substituting values and possibly using numerical approximation. In this exercise, after substituting \( \pi \) into the expression and simplifying, we arrive at approximate values for the numerator and denominator.

Dividing the simplified results \( 24.2944 \) by \( 40.0648 \) gives an approximate limit value of \( 0.6061 \). This step involves simple division and helps to finalize the limit evaluation process. By breaking down the problem into smaller parts—substitute, simplify, then divide—students can systematically arrive at the limit’s approximate value. This approach not only ensures accuracy but enhances understanding of the fundamental processes involved in limit evaluation.