Problem 88 - 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 the function \(f(x^2)\) as x approaches -1 is 4.

Step 1: Recall the properties of limits

To be able to solve this exercise easily, we should first recall some important properties of limits, such as the substitution rule, which states: If \(\lim_{x \rightarrow a} g(x) = L\), then \(\lim_{x \rightarrow a} f(g(x)) = f(L)\). We will be using this property to find the desired limit.

Step 2: Use the given limit information

We are given that \(\lim_{x \rightarrow 1} f(x) = 4\). This means that as x approaches 1, the function f(x) approaches the value 4.

Step 3: Find the limit as x approaches -1 of x^2

Since we want to find \(\lim_{x \rightarrow -1} f(x^2)\), let's first find the limit of x^2 as x approaches -1. Using the direct substitution method, we get: $$\lim_{x \rightarrow -1} x^2 = (-1)^2 = 1$$

Step 4: Apply the substitution rule for limits

Now that we have found the limit of x^2 as x approaches -1, we can use the substitution rule to find the desired limit. We know \(\lim_{x \rightarrow -1} x^2 = 1\), and we are given that \(\lim_{x \rightarrow 1} f(x) = 4\). Hence, we can substitute the limit of x^2 into the function f(x): $$\lim_{x \rightarrow -1} f(x^2) = f\left(\lim_{x \rightarrow -1} x^2\right) = f(1)$$ Since we know that \(\lim_{x \rightarrow 1} f(x) = 4\), it follows that \(f(1) = 4\). Therefore, the desired limit is: $$\lim_{x \rightarrow -1} f(x^2) = f(1) = \boxed{4}$$