Question

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

Short answer

Answer: The limit of the function $f(x^2)$ as x approaches -1 is 4.

Step-by-step solution

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 $\underset{x\to a}{lim}g(x)=L$, then $\underset{x\to a}{lim}f(g(x))=f(L)$. We will be using this property to find the desired limit.

Use the given limit information

We are given that $\underset{x\to 1}{lim}f(x)=4$. This means that as x approaches 1, the function f(x) approaches the value 4.

Find the limit as x approaches -1 of x^2

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

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 $\underset{x\to -1}{lim}{x}^2=1$, and we are given that $\underset{x\to 1}{lim}f(x)=4$. Hence, we can substitute the limit of x^2 into the function f(x): $$\underset{x\to -1}{lim}f(x^2)=f\left(\underset{x\to -1}{lim}{x}^2\right)=f(1)$$ Since we know that $\underset{x\to 1}{lim}f(x)=4$, it follows that $f(1)=4$. Therefore, the desired limit is: $$\underset{x\to -1}{lim}f(x^2)=f(1)=\boxed{{\displaystyle 4}}$$