In our previous article, we discussed some activities to teach students how to find the limit at a vertical asymptote of a rational function. In this article, we will continue with additional activities that can help students understand this concept better.

**1. Graphical Approach:**

Draw the graph of a rational function with vertical asymptotes. Then, ask the students to identify the x-coordinates of the vertical asymptotes. Once they have identified all the asymptotes, have them plot points on both sides of the vertical asymptote and calculate their y-coordinates. Finally, ask them to draw a line that represents the limit of the function as x approaches the vertical asymptote.

**2. Analytical Approach:**

Give the students a rational function with a vertical asymptote. Write the expression for the function as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials with the same degree and the leading coefficient of Q(x) is not equal to zero. Then, explain that to find the limit at the vertical asymptote, they need to find the limit of the function as x approaches the value of the vertical asymptote from both sides.

For example, consider the rational function f(x) = (x+2)/(x^2 – 1). The vertical asymptotes of this function are x = 1 and x = -1. To find the limit at x = 1, we need to evaluate the following limits:

Limit as x approaches 1 from the left: lim f(x) = lim (x+2)/(x^2 – 1) = -∞

x→1^(-)

Limit as x approaches 1 from the right: lim f(x) = lim (x+2)/(x^2 – 1) = ∞

x→1^(+)

Therefore, the limit at x = 1 does not exist.

**3. Real-Life Examples:**

Use real-life examples to illustrate the concept of vertical asymptotes. For example, you can ask the students to imagine a roller coaster that drops very steeply. The drop represents the vertical asymptote, and the limit represents how close the riders get to the ground. Ask the students to describe what happens to the riders as they approach the vertical drop and how close they get to the ground.

**4. Group Activities:**

Divide the class into groups and give each group a different rational function with a vertical asymptote. Then, have them find the limit at the vertical asymptote using different methods, such as graphical approach, analytical approach, or real-life examples. Finally, ask them to present their findings to the class and identify the similarities and differences in their approaches.

In conclusion, these activities can be helpful in teaching students how to find the limit at a vertical asymptote of a rational function. By using real-life examples and encouraging group discussions, students can better understand this concept and develop their problem-solving skills.