Questions: Determine whether the following statements are true and give an explanation or counterexample. d. If lim(x -> a) f(x) = ∞ or lim(x -> a) f(x) = -∞, then lim(x -> a) f(x) does not exist. Is this statement true?

Solution Steps

To determine whether the statement is true, we need to understand the definition of limits involving infinity. Specifically, if the limit of a function as \( x \) approaches a certain value \( a \) is either \( \infty \) or \( -\infty \), it means the function grows without bound in the positive or negative direction, respectively. According to the formal definition of limits, if a function approaches \( \infty \) or \( -\infty \) as \( x \) approaches \( a \), the limit does not exist in the finite sense, but it does exist in the extended sense as an infinite limit.

1. Understand the definition of limits involving infinity.
2. Check if the limit approaching \( \infty \) or \( -\infty \) means the limit does not exist in the finite sense.
3. Conclude whether the statement is true based on the definition.

To determine whether the statement is true, we need to understand the definition of limits involving infinity. Specifically, if the limit of a function as \( x \) approaches a certain value \( a \) is either \( \infty \) or \( -\infty \), it means the function grows without bound in the positive or negative direction, respectively.

According to the formal definition of limits, if a function approaches \( \infty \) or \( -\infty \) as \( x \) approaches \( a \), the limit does not exist in the finite sense. However, it does exist in the extended sense as an infinite limit.

Given the definition, the statement "If \( \lim_{x \rightarrow a} f(x) = \infty \) or \( \lim_{x \rightarrow a} f(x) = -\infty \), then \( \lim_{x \rightarrow a} f(x) \) does not exist" is true because the limit does not exist in the finite sense.

Final Answer