Problem 49 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to 5}\ (10-x^{2})$$

Step-by-Step Solution

Answer

The limit of the function \( (10-x^{2}) \) as \( x \) approaches 5 is -15.

Step 1: Understand the Problem

We are given a function \( f(x) = 10 - x^{2} \) and we need to find the limit of this function as \( x \) approaches 5. This is best done by direct substitution, which involves substituting the limit value directly into the function.

Step 2: Direct Substitution

Replace \( x \) in the function \( f(x) = 10 - x^{2} \) with the limit value 5. So the function becomes \( f(5) = 10 - 5^{2} \).

Step 3: Compute the Limit

Now, evaluate \( f(5) \). This gives \( f(5) = 10 - 25 = -15 \). Hence, the limit of the function \( (10-x^{2}) \) as \( x \) approaches 5 is -15.

Key Concepts

Direct Substitution

Direct substitution is a method used to find the limit of a function when the function is continuous at a point. In this approach, you simply replace the variable with the value it is approaching.
For example, if you need to find \( \lim_{x \to 5} (10-x^2) \), you replace \( x \) with 5.
This is simple because the function \( 10 - x^2 \) is a continuous polynomial.

• Start by ensuring the function is not undefined at the point you are substituting.
• If it's defined, substitute the value directly.

Direct substitution works best with polynomial, exponential, or trigonometric functions unless there's a discontinuity or division by zero. This method gives a quick solution to the problem without needs for complex calculus techniques.

Evaluating Functions

Evaluating a function means finding the value of the function for a specific input. In the context of limits, you want to evaluate the function near the point that \( x \) is approaching.
Consider the function \( f(x) = 10 - x^2 \). If you are asked to find \( f(5) \), you simply substitute 5 into the equation:
\[ f(5) = 10 - 5^2 = 10 - 25 = -15 \]

• Ensure the function value at a specific point is within its domain.
• Plug in the values precisely to avoid calculation errors.

Evaluating functions this way helps confirm that they behave as expected at particular points, and is especially useful in practical applications like finding limits.

Limit of a Function

The concept of a limit is fundamental to calculus. It describes the behavior of a function as the input approaches a certain value. The notation \( \lim_{x \to c} f(x) \) denotes the limit of \( f(x) \) as \( x \) approaches \( c \).
When we say \( \lim_{x \to 5} (10-x^2) = -15 \), it implies that as \( x \) gets closer and closer to 5, the values of the function \( 10-x^2 \) approach -15.

• Limits help understand how functions behave at singularities where they might not be directly evaluated.
• They are crucial in defining derivatives and integrals.

Understanding limits allows you to make sense of how a function progresses towards certain values, providing a foundation for more advanced calculus concepts.