91Ó°ÊÓ

First, we need to identify the numerator and denominator of the function. In this case, we have: - Numerator: \(x^3\) - Denominator: \(x^2\)

Now, we need to find the limits of the numerator and denominator as \(x \rightarrow 0\). - Limit of the numerator: $$ \lim_{x \rightarrow 0} x^3 = 0 $$ - Limit of the denominator: $$ \lim_{x \rightarrow 0} x^2 = 0 $$

Now, we'll rewrite the function in terms of the limit of the numerator and denominator. $$ \lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} \frac{x^3}{x^2} $$

To find the limit of the function, it's often helpful to simplify the function first. In this case, by canceling out one of the x's. $$ \lim_{x \rightarrow 0} \frac{x^3}{x^2} = \lim_{x \rightarrow 0} \frac{x^3}{x^2} \cdot \frac{1/x}{1/x} = \lim_{x \rightarrow 0} x $$ Taking the limit as x approaches 0: $$ \lim_{x \rightarrow 0} x = 0 $$ So the limit of the function is \(0\). However, during the process we find out that the Numerator and Denominator both goes to \(0\) as \(x\rightarrow0\). This function is not giving us the desired limit of the form \(\infty / \infty\). We have to try another function Consider the function: $$ g(x) = \frac{1}{x} \cdot \frac{x}{1} $$ Following the same steps as before, we have: - Numerator: \(x\) - Denominator: \(x\) - Limit of the numerator: $$ \lim_{x \rightarrow 0} x = 0 $$ - Limit of the denominator: $$ \lim_{x \rightarrow 0} x = 0 $$ - Rewrite the function in terms of the limit: $$ \lim_{x \rightarrow 0} g(x) = \lim_{x \rightarrow 0} \frac{x}{x} $$ Now the limit of the denominator and numerator both goes to infinity as x approaches 0, so we have: $$ \lim_{x \rightarrow 0} \frac{x}{x} = \frac{\infty}{\infty} $$ Therefore, \(g(x) = \frac{x}{x}\) is an example of a limit of the form \(\infty / \infty\) as \(x \rightarrow 0\).