Question

Graphical, Numerical, and Analytic Analysis, use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 16} \frac{4-\sqrt{x}}{x-16} $$

Short answer

The value of the limit as x approaches 16 for the given function, as estimated by graphical, numerical, and analytic methods, is 0.

Step-by-step solution

Graphing the Function

Plot the function \( f(x) = \frac{4 - \sqrt{x}}{x - 16} \) using a graphing software. Since we are interested in the limit as x approaches 16, focus on the behavior of the function around x = 16. Notice the hole in the graph at x = 16 -- it's indicative of a limit.

Numerical Analysis

To analyze numerically, generate a table of values. Choose values of x that are close to 16 (both lesser and greater) and calculate the corresponding y-values using the function \( f(x) = \frac{4 - \sqrt{x}}{x - 16} \). Observe the behavior of the y-values as x approaches 16.

Analytical Methods

For the analytical method, an algebraic manipulation can be done: Multiply the numerator and denominator by the conjugate of the numerator. So, the function will be transformed into \( \lim_{x \rightarrow 16} \frac{(4 - \sqrt{x})*(4 + \sqrt{x})}{(x - 16)*(4 + \sqrt{x})} \). Now the (x-16) terms cancel out and the limit is easy to compute: \( \lim_{x \rightarrow 16}(4-\sqrt{x}) = 4-\sqrt{16} = 0 \)