Describe the behavior of the function near the vertical asymptote.
Is there another way to word this problem so that I might understand?
Seriously, I have gotten everything so far, so is this just worded weird?
Certainly! Let's rephrase the question to ensure clarity:
What is the behavior of the function as it approaches the vertical asymptote?
This question is asking about how the function behaves or what happens to the function as it gets closer and closer to the vertical asymptote.
Now, let's explore how to determine the behavior of a function near a vertical asymptote:
1. Identify the vertical asymptote(s) of the function. Vertical asymptotes occur when the value of the function approaches infinity or negative infinity as the input approaches a certain value.
2. Determine the function's behavior as it approaches the vertical asymptote from both sides. You can do this by taking the limit of the function as the input approaches the vertical asymptote value from the left (negative side) and from the right (positive side). Evaluate these two limits separately.
a. If both limits approach the same value (either positive or negative infinity), then the function behaves similarly from both sides and has a "straight" or "smooth" behavior near the vertical asymptote.
b. If the limits approach different values (one is positive infinity and the other is negative infinity), then the function has a "jump" or "discontinuity" at the vertical asymptote. This indicates a vertical asymptote with a "hole" in the graph.
c. If the limits do not exist (meaning they approach different values with no convergence to infinity), then the function has a "vertical asymptote with a vertical tangent line" or a "vertical cusp" at the asymptote. This suggests that the function approaches infinity or negative infinity rapidly near the vertical asymptote.
By considering these steps, you can describe the behavior of a function near its vertical asymptote.