Problem 18 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow-5} \frac{5}{x+3} $$

Step-by-Step Solution

Answer

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

Step 1: Substituting x

First, substitute \(x = -5\) into the expression \(\frac{5}{x+3}\): \(\frac{5}{-5+3}\)

Step 2: Simplify the denominator

Now simplify the denominator: \(\frac{5}{-2}\)

Step 3: Divide

Finally, simplify the whole fraction: \(\frac{-5}{2}\)

Key Concepts

Limit of a Function

When we talk about the limit of a function in calculus, we're referring to the behavior of the function as the input approaches a certain value. To find the limit, we want to see what value the function is approaching, not necessarily what the function equals at that point.

For example, when asked to find \(\lim _{x \rightarrow-5} \frac{5}{x+3}\), we are investigating what happens to \(\frac{5}{x+3}\) as 'x' gets very close to -5. It's important to note that we are not substituting 'x' for -5 to find the function's value at that point but rather its behavior around that point. Limit calculations can range from the straightforward, like direct application and evaluation, to the complex, involving rules and techniques to handle forms that are not readily solvable by direct substitution.

Understanding limits is crucial, as they are foundational in calculus, particularly in understanding continuity, defining derivatives, and calculating integrals.

Substitution in Limits

The technique of substitution in limits is often the first step attempted when finding a limit. This method involves directly replacing the variable in the expression with the value it's approaching. If this substitution gives us a definite number, then that number is the limit.

For the limit problem \(\lim _{x \rightarrow-5} \frac{5}{x+3}\), the first step is to do a direct substitution, which means replacing 'x' with -5. This gives us the expression \(\frac{5}{-5+3}\), which simplifies to \(\frac{5}{-2}\). In cases where substitution leads to an indeterminate form - such as 0/0 or \infty/\infty - further techniques and limit laws are employed to find the limit. However, if substitution yields a real number, the process is completed, and the limit is found.

Simplifying Expressions

The process of simplifying expressions is a key skill in calculus as it makes handling complex equations more manageable. Simplifying can involve reducing fractions, combining like terms, or using algebraic identities to rewrite expressions in a more straightforward form.

In our example, once we substitute 'x' with -5, we simplify the expression by evaluating the denominator -5+3, which reduces to -2. Then, we simplify the fraction \(\frac{5}{-5+3}\) to \(\frac{5}{-2}\), or -2.5. Simplifying expressions helps in the next steps of solving limits and ensures that the problem is as easy to work with as possible. It's essential to be thorough with simplification to avoid mistakes in subsequent calculations and conclusions.