Problem 104 - Calculus: Early Transcendentals

Question

If \(\lim _{x \rightarrow 1} f(x)=4,\) find \(\lim _{x \rightarrow-1} f\left(x^{2}\right)\).

Step-by-Step Solution

Answer

Answer: The limit of f(x²) as x approaches -1 is 4.

Step 1: Substitute x² for x

First, we need to substitute x² for x in the limit expression: \(\lim_{x\rightarrow -1} f(x^2)\)

Step 2: Observe the behavior of x² as x approaches -1

Notice that as \(x \rightarrow -1\), \(x^2 \rightarrow 1\). This is because \((-1)^2 = 1\). We will use this fact to rewrite the limit expression.

Step 3: Rewrite the limit expression

Since \(x^2 \rightarrow 1\) as \(x \rightarrow -1\), we can rewrite the limit expression as \(\lim_{x^2 \rightarrow 1} f(x^2)\)

Step 4: Utilize the given limit

We are given that \(\lim_{x \rightarrow 1} f(x) = 4\). Since \(x^2 \rightarrow 1\), we can use the given limit to find the limit of \(f(x^2)\) as \(x^2 \rightarrow 1\).

Step 5: Evaluate the limit

Using the given limit, we have \(\lim_{x^2 \rightarrow 1} f(x^2) = f(1) = 4\). Therefore, the limit of \(f(x^2)\) as \(x\) approaches -1 is 4.

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial in calculus as it deals with the behavior of a function as the input approaches a particular value. It is not always about where the function is at that exact value, but rather where it is 'heading'.

For example, looking at our exercise, when we say \( \lim_{x \rightarrow 1} f(x) = 4 \), we are exploring what value the function \(f(x)\) is approaching as \(x\) gets closer and closer to 1, but not necessarily touching it. In the realm of mathematical analysis, it's like predicting where a particle will end up as it approaches a certain point without actually reaching it. As we're dealing with \(f(x^2)\), we are examining how the function behaves as the square of \(x\) moves towards a particular value.

Substitution Method

When we come across a situation where direct evaluation of the limit is not possible, the substitution method can be particularly useful. This method involves replacing a part of our function with a different expression that behaves similarly near the point of interest.

In our step-by-step example, as \(x\) approaches -1, note that \(x^2\) naturally approaches 1. Hence, we substitute \(x^2\) with 1 within our limit expression. This simplifies our task, as we utilize the given limit of \(f(x)\) at 1, avoiding the direct substitution of -1 into \(x\), which isn't helpful in this case. It's like switching out a complicated part for a simpler one to make our analysis manageable.

Properties of Limits

The properties of limits are sets of rules that make finding limits a more systematic process. Some of these rules include the limit of a constant function, the limit of a sum, product, and quotient of functions, as well as the power rule.

These properties greatly simplify calculating limits for more complex functions. In our example, knowing that the limit of a function as \(x\) approaches a value is the same as the limit of that function as \(x^2\) approaches the square of that value (under some conditions) is one such property. It stems from the predictable behavior of continuous functions—they follow certain paths that can be anticipated through these properties. By internalizing these properties, students can easily navigate through a wide variety of limit problems with confidence.