The image presents a mathematical question in Indonesian that can be translated as follows:

Question: Suppose the function $f$ satisfies the following limit condition: $\lim_{{x \to \infty}} f(x) = \infty.$ Does this imply that the function $f$ has both a vertical asymptote and a horizontal asymptote?

The answer choices are:

• True
• False

Explanation:

• Vertical Asymptote: A vertical asymptote occurs when the function approaches infinity as $x$ approaches a finite value, not infinity. Therefore, the condition $\lim_{{x \to \infty}} f(x) = \infty$ does not directly indicate the presence of a vertical asymptote.
• Horizontal Asymptote: A horizontal asymptote exists if $\lim_{{x \to \infty}} f(x)$ approaches a finite number. In this case, since the limit goes to infinity, there is no horizontal asymptote either.

Hence, the correct answer is False. The given condition does not imply the existence of either a vertical or horizontal asymptote.

Would you like further clarification or more detailed information on limits and asymptotes?

Here are some additional related questions:

1. What is the formal definition of a vertical asymptote?
2. Can a function with an infinite limit at infinity still have a horizontal asymptote?
3. How do asymptotes differ between rational and exponential functions?
4. What is the significance of slant (oblique) asymptotes?
5. How does the behavior of a function near infinity help in identifying its asymptotes?

Tip: Always carefully examine the behavior of the function both at finite and infinite points to identify the types of asymptotes it may have.