To verify the vertical asymptote of the function $f(x) = \frac{x - 2}{x}$ using limits, we can analyze the behavior of $f(x)$ as $x$ approaches the suspected vertical asymptote, which is $x = 0$.

We need to evaluate the left-hand limit and the right-hand limit as $x$ approaches 0:

1. Right-Hand Limit ($x$ approaches 0 from the right):

This is the limit as $x$ approaches 0 from the positive side ($x \to 0^+$).

$\lim_{x \to 0^+} \frac{x - 2}{x} = \lim_{x \to 0^+} \left(\frac{x}{x} - \frac{2}{x}\right) = \lim_{x \to 0^+} \left(1 - \frac{2}{x}\right)$

As $x$ approaches 0 from the right, $\frac{2}{x}$ becomes very large (positive), so:

$\lim_{x \to 0^+} \left(1 - \frac{2}{x}\right) = 1 - (+\infty) = -\infty$

2. Left-Hand Limit ($x$ approaches 0 from the left):

This is the limit as $x$ approaches 0 from the negative side ($x \to 0^-$).

$\lim_{x \to 0^-} \frac{x - 2}{x} = \lim_{x \to 0^-} \left(\frac{x}{x} - \frac{2}{x}\right) = \lim_{x \to 0^-} \left(1 - \frac{2}{x}\right)$

As $x$ approaches 0 from the left, $\frac{2}{x}$ becomes very large (negative), so:

$\lim_{x \to 0^-} \left(1 - \frac{2}{x}\right) = 1 - (-\infty) = +\infty$

Conclusion:

Since the left-hand limit and right-hand limit as $x$ approaches 0 are $+\infty$ and $-\infty$ respectively, this confirms that $x = 0$ is indeed a vertical asymptote of the function $f(x) = \frac{x - 2}{x}$.

Would you like to dive deeper into any part of this solution?

Here are some related questions:

1. How do the limits help in identifying horizontal asymptotes?
2. What is the significance of different signs in left-hand and right-hand limits at the asymptote?
3. Can a function have more than one vertical asymptote? How?
4. How does the vertical asymptote affect the graph of the function near $x = 0$?
5. What is the end behavior of $f(x)$ as $x$ approaches $\infty$ or $-\infty$?

Tip: When verifying asymptotes using limits, it's essential to check both the left-hand and right-hand limits, as they can provide different insights into the behavior of the function near the asymptote.