Problem 50 - Precalculus with Limits

Question

GRAPHICAL, NUMERICAL, AND ALGEBRAIC ANALYSIS In Exercises 49-54, (a) graphically approximate the limit (if it exists) by using a graphing utility to graph the function, (b) numerically approximate the limit (if it exists) by using the \(table\) feature of a graphing utility to create a table, and (c) algebraically evaluate the limit (if it exists) by the appropriate technique(s). $$\lim_{x \to 5^+} \dfrac{5-x}{25-x^2}$$

Step-by-Step Solution

Answer

The exact value of limit depends on the results of the graphical, table, and algebraic computations. The limit should be the same for all three methods, if it exists since these are just different methods to approximate the limit of the function.

Step 1: Graphical Limit

Graph the function \( f(x)=\frac{5-x}{25-x^2} \) using a graphing utility. As x approaches 5 from the right, observe the value of the function. The y-value the function approaches is the limit. The closer to point 5 (on the x-axis) the more accurate limit approximation.

Step 2: Numerical Limit

Use the table feature of the graphing utility to construct a table of values for the function as x approaches 5 from the right. Observe the value to which the function's value is tending. Specifically, use values like 5.1, 5.01, 5.001, etc. The value \( f(x) \) approaches as \( x \) gets closer and closer to 5 from the right is the numerical limit.

Step 3: Algebraic Limit

To evaluate limit algebraically, first substitute x=5 in the function \( f(x)=\frac{5-x}{25-x^2} \). If the result is indeterminate form i.e., 0/0 or Infinity/Infinity, use l'Hopital's rule. According to l'Hopital’s rule, the limit of the quotient of two functions is equal to the limit of the quotient of their derivatives, whenever the original limit is of indeterminate form. Differentiate the numerator and denominator separately, and then substitute x=5 into the new fraction to get the limit.

Key Concepts

Graphical Limit

When we talk about a graphical limit, we are looking to visually determine the behavior of a function as it approaches a certain point, in this case, when \( x \to 5^+ \). To do this:

• Use a graphing utility to plot the function \( f(x) = \frac{5-x}{25-x^2} \).
• Focus on the curve’s behavior as it approaches \( x = 5 \) from the right-hand side. This is indicated by the \( x \to 5^+ \) notation.

As \( x \) gets closer to 5 from the positive side, observe the \( y \)-value where the function heads. This \( y \)-value will give an approximation of the limit. If your graph is accurate and you zoom adequately around \( x = 5 \), you should notice a trend in the function's behavior. This trend is crucial in approximating the limit graphically.

Numerical Limit

Numerical limits involve approaching a solution using calculated values. By inputting values that are close to our point of interest into the function \( f(x) = \frac{5-x}{25-x^2} \), we can observe patterns in the output. To effectively find a numerical limit:

• Utilize the table feature of a graphing utility to calculate the function's value as \( x \) approaches 5 from larger values, such as 5.1, 5.01, and 5.001.
• Examine how these values change and tend toward a specific number.

Through this method, the output or \( y \)-values should begin to stabilize or approach a certain value as \( x \) becomes closer to 5 from the right. This consistent \( y \)-value is what we describe as the numerical approximation of the limit.

Algebraic Limit Evaluation

The algebraic method allows us to analytically determine the limit of a function. When given a function like \( f(x) = \frac{5-x}{25-x^2} \), direct substitution of \( x = 5 \) typically tests if the function results in a meaningful number or an indeterminate form like \( \frac{0}{0} \). If the result is indeterminate:

• Apply l’Hôpital's Rule, which states that the limit of the ratio of two functions is the same as the limit of the ratio of their derivatives, provided the original limit yields an indeterminate form.
• Find the derivatives separately for the numerator \( 5-x \) and the denominator \( 25-x^2 \), and take their ratio.
• Substitute \( x = 5 \) again in this new function to find the limit.

This method is robust and allows us to systematically eliminate indeterminate forms to find a precise limit value.