Problem 54 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to -5}\ \dfrac{6}{x+2}$$

Step-by-Step Solution

Answer

The limit of \(\frac{6}{x+2}\) as x approaches -5 is -2.

Step 1: Substitute the Limiting Value

Substitute -5 for x in the function \(\frac{6}{x+2}\). This will give: \[ \frac{6}{-5+2}\].

Step 2: Simplify the Expression

Simplify \( \frac{6}{-5+2}\) to get \( \frac{6}{-3}\).

Step 3: Find the Quotient

Find the quotient of \( \frac{6}{-3}\) which gives -2.

Key Concepts

Direct Substitution

When faced with a problem involving limits, the first thing to try is the method of direct substitution. This means you replace the variable in the function with the limit value directly. For instance, when trying to find the limit of the function \( \lim_{x \to -5} \frac{6}{x+2} \), you start by substituting \(-5\) for \(x\).
This gives us the expression \( \frac{6}{-5+2} \).

• Direct substitution is often the easiest method.
• If it results in a well-defined number, you're done!
• If not, other methods like simplification might be necessary.

Using direct substitution helps us move quickly in many cases, especially when it results in a clean and simple number.

Simplifying Expressions

The next step after direct substitution is to simplify the expression to its simplest form. In this exercise, the expression \( \frac{6}{-5+2} \) needs simplifying.
Here's how you do it:

• Calculate \(-5 + 2\), which equals \(-3\).
• Substitute this back into the expression, giving us \( \frac{6}{-3} \).
• This simplifies the fraction to a more understandable form.

Simplification is all about making numbers easier to work with and to better understand the result of your calculations.
By simplifying, you often get a clearer view of the behavior of the function as \(x\) approaches the limit value.
Additionally, this step ensures that you're working with the correct numbers as you proceed to finding the quotient.

Quotients

Once you have your simplified expression, finding the quotient is the final step.
In this exercise, after simplification, we have the expression \( \frac{6}{-3} \).

• To find the quotient, divide \(6\) by \(-3\).
• This gives you a result of \(-2\).

Quotients are about dividing one number by another. It often represents the final solution you are seeking in limit problems.
In this context, obtaining the quotient completes the limit calculation, showing us the behavior of the function \( \lim_{x \to -5} \frac{6}{x+2} \) approaches \(-2\) as \(x\) nears \(-5\).
This method is straightforward but fundamental because it reveals how direct substitution and simplification contribute to understanding the function's limits.
Hence, practicing this process can make more complex limits seem less daunting.