To evaluate the limit

$\lim_{x \to 0^{-}} \frac{3}{x^2},$

we will proceed as follows:

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Step 1: Understanding the function behavior

The given function is

$f(x) = \frac{3}{x^2}.$

For $x^2 > 0$, the denominator is always positive, regardless of whether $x$ is positive or negative. Thus, $f(x)$ is always positive for any $x \neq 0$.

1. As $x$ approaches 0 from the left ($x \to 0^{-}$):
   • The denominator $x^2$ becomes very small, approaching 0.
   • The fraction $\frac{3}{x^2}$ grows without bound, approaching infinity.

This suggests: $\lim_{x \to 0^{-}} \frac{3}{x^2} = \infty.$

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Step 2: Confirming with a table of values

We can substitute values of $x$ approaching 0 from the left ($x \to 0^{-}$) and observe the trend of $f(x)$:

$x$	$-0.1$	$-0.01$	$-0.001$	$-0.0001$
$f(x)$	$300$	$30,000$	$3,000,000$	$30,000,000$

As $x \to 0^{-}$, $f(x)$ grows larger and larger, confirming the result.

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Step 3: Graphing the function

The graph of $f(x) = \frac{3}{x^2}$ is symmetric about the $y$-axis because it depends on $x^2$. It shows vertical asymptotes at $x = 0$ and decreases towards 0 as $|x|$ increases.

Let me generate a graph for clarity. numpy as np import matplotlib.pyplot as plt

Define the function f(x) = 3 / x^2

def f(x): return 3 / (x**2)

Generate x values excluding zero (approaching zero from both sides)

x_left = np.linspace(-0.2, -0.001, 500) # x approaching 0 from the left x_right = np.linspace(0.001, 0.2, 500) # x approaching 0 from the right

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x_left, f(x_left), label=r"$f(x) = \frac{3}{x^2}$ (Left)", color="blue") plt.plot(x_right, f(x_right), label=r"$f(x) = \frac{3}{x^2}$ (Right)", color="red")

Highlight the asymptote at x = 0

plt.axvline(0, color="black", linestyle="--", label="Vertical Asymptote at $x=0$")

Add labels and legend

plt.title("Graph of $f(x) = \frac{3}{x^2}$") plt.xlabel("$x$") plt.ylabel("$f(x)$") plt.ylim(0, 100) plt.legend() plt.grid() plt.show()The graph confirms the behavior of $f(x) = \frac{3}{x^2}$:

1. As $x \to 0^{-}$, the function value grows very large ($f(x) \to \infty$).
2. The vertical asymptote at $x = 0$ is evident.

Thus, the limit is: $\lim_{x \to 0^{-}} \frac{3}{x^2} = \infty.$

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Do you want further details or have any questions?

Here are related queries to explore:

1. What is $\lim_{x \to 0^{+}} \frac{3}{x^2}$?
2. How does the behavior of $f(x)$ change for negative and positive values of $x$?
3. Can we compute the one-sided limits numerically using additional smaller values of $x$?
4. What does the symmetry of $f(x) = \frac{3}{x^2}$ about the $y$-axis imply?
5. What happens to the function as $x \to \pm \infty$?

Tip: Always verify limits involving infinity with numerical approximations and graphs for clarity!