To find a vertical asymptote of a function using limits, you generally need to check where the function approaches infinity or negative infinity as $x$ approaches certain values. Here's the step-by-step process:

Steps to find vertical asymptotes using limits:

1. Identify possible vertical asymptotes:

   • Vertical asymptotes typically occur where the function is undefined, often at values of $x$ that make the denominator of a rational function equal to zero.
   • For example, if you have $f(x) = \frac{P(x)}{Q(x)}$, find the values of $x$ that make $Q(x) = 0$.

2. Take the limit of the function as $x$ approaches the suspected values (from step 1) from both the left ($x \to a^-$) and the right ($x \to a^+$):

   • $\lim_{x \to a^-} f(x)$
   • $\lim_{x \to a^+} f(x)$

   If either of these limits tends to $+\infty$ or $-\infty$, then the line $x = a$ is a vertical asymptote.

3. Conclusion:

   • If the function tends to $\infty$ or $-\infty$ as $x$ approaches $a$, you have a vertical asymptote at $x = a$.

Example:

Find the vertical asymptote of $f(x) = \frac{1}{x-2}$.

1. The function is undefined at $x = 2$ (since the denominator becomes 0), so we suspect $x = 2$ is a vertical asymptote.

2. Compute the limit as $x \to 2^-$ and $x \to 2^+$: \lim_{x \to 2^-} \frac{1}{x - 2} = -\infty \quad \text{(as x approaches 2 from the left)} \lim_{x \to 2^+} \frac{1}{x - 2} = +\infty \quad \text{(as x approaches 2 from the right)}

3. Since the limits approach $\pm \infty$, there is a vertical asymptote at $x = 2$.

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Would you like more details or examples? Here are five related questions for further exploration:

1. How can you distinguish between vertical and horizontal asymptotes using limits?
2. What happens if a function approaches a finite value at a point where the denominator is zero?
3. How do you find asymptotes for functions involving square roots?
4. Can polynomials have vertical asymptotes, and how would you find them?
5. How do vertical asymptotes differ from holes in the graph?

Tip: Always check the one-sided limits to confirm the direction (positive or negative infinity) when determining vertical asymptotes.