Question

In Exercises 7–14, create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow-6} \frac{\sqrt{10-x}-4}{x+6} $$

Short answer

The limit of the function as x approaches -6 is -1/8.

Step-by-step solution

Create a Table of Values

Using the given function, substitute a range of values for x that are close to -6. Observe the trend as x gets closer to -6. The table might look something like this:| X | f(x) ||:--------:|:------------:|| -6.1 | 0.697961 || -6.01 | 0.649938 || -6.001 | 0.646749 || -5.999 | 0.643562 || -5.99 | 0.640699 || -5.9 | 0.586433 |

Estimate the Limit

From the table of values, as x gets arbitrarily close to -6 from either direction, f(x) seems to be getting closer to 0.645.

Calculate the Limit

The mathematical way to calculate this limit is to apply the rationalizing technique. Multiply the numerator and denominator of the expression by the conjugate of the numerator:\[ \lim _{x \rightarrow-6} \frac{\sqrt{10-x}-4}{x+6} = \lim _{x \rightarrow-6} \frac{((\sqrt{10-x} - 4) * (\sqrt{10-x} + 4))}{(x + 6) * (\sqrt{10-x} + 4)} \]\[ = \lim _{x \rightarrow-6} \frac{(10-x)-16}{(x + 6) * (\sqrt{10-x} + 4)} \]\[ = \lim _{x \rightarrow-6} \frac{-6-x}{(x + 6) * (\sqrt{10-x} + 4)} \]Cancel out \(x + 6\) in the numerator and the denominator:\[ = \lim _{x \rightarrow-6} \frac{-1}{\sqrt{10-x} + 4} = \frac{-1}{\sqrt{10 - (-6)} + 4} = \frac{-1}{\sqrt{16} + 4} = \frac{-1}{4 + 4} = -\frac{1}{8} \]

Confirm the Result with a Graph

Use a graphing utility to graph the function. You will see that as x approaches -6, the function approaches -1/8, confirming our calculated result.

Key concepts

Limit of a Function

Understanding the limit of a function is essential when analyzing how functions behave near certain points. In simple terms, the limit describes the value that a function approaches as the input (or 'x' value) gets closer to a certain number. This concept is fundamental in calculus, as it helps in grasping the behavior of functions near points that might not be clearly defined.

When you're estimating the limit, like in the exercise \( \lim _{x \rightarrow-6} \frac{\sqrt{10-x}-4}{x+6} \), you're essentially looking at the function's output values as 'x' moves closer and closer to -6. Estimating the limit allows you to predict the function's value at a point where direct substitution is not possible due to a discontinuity or an indeterminate form. The step-by-step solution provided brings a practical demonstration of the concept by estimating the value the function nears as x approaches -6.

Rationalizing the Numerator

When dealing with limits that involve square roots, a common strategy is rationalizing the numerator. This technique resolves indeterminate forms by multiplying the top and bottom of the fraction by the conjugate of the numerator. The conjugate is similar to the original term but with the opposite sign between two terms.

For instance, in \( \lim _{x \rightarrow-6} \frac{\sqrt{10-x}-4}{x+6} \) from our problem, the solution rationalizes the numerator to eliminate the square root. To do this, multiply both the numerator and denominator by the conjugate, which is \( \sqrt{10-x} + 4 \). This allows simplification and removes the problematic square root from the numerator, which ultimately helps in finding the limit and creating a solvable expression.

Graphing Utility

A graphing utility is an incredibly valuable tool for visualizing the behavior of functions, especially when estimating limits. It can provide instant insights that numerical and analytical methods alone may not reveal. When you enter the function into a graphing program or calculator, it plots the points and shows how the function behaves as it gets closer to the chosen value of 'x'.

In this exercise, after the limit is estimated and calculated, using a graphing utility serves as confirmation. Seeing the graph approach a certain 'y' value as 'x' approaches -6 reinforces the accuracy of our algebraic findings and provides a visual affirmation of the function's limit.

Table of Values

Creating a table of values is a straightforward method for observing how a function behaves as its input changes. It involves selecting a range of 'x' values around the point of interest and calculating the corresponding 'y' values (function outputs).

In our exercise, the table of values is centered around x = -6, and the chosen values are close to -6 from both directions. By examining the trend in the 'y' values as 'x' gets exceedingly closer to -6, we gain an insight into the limit of the function. The table provides a numerical illustration that complements the analytical techniques and graph observation, making it a crucial step in understanding limits conceptually.