Let's tackle these two questions using the epsilon-delta definition of limits. I'll first give a brief overview of how the epsilon-delta definition works, and then we'll address the true/false statements.

Epsilon-Delta Definition of Limit:

For a function $f(x)$, we say that: $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $0 < |x - a| < \delta$, we have $|f(x) - L| < \epsilon$.

Question (a):

Statement: IF $\lim_{x \to a} [f(x) \cdot g(x)]$ exists and $\lim_{x \to a} g(x)$ exists, THEN $\lim_{x \to a} f(x)$ exists.

Let's break it down:

• Let $L_1 = \lim_{x \to a} [f(x) \cdot g(x)]$ and $L_2 = \lim_{x \to a} g(x)$.
• We need to show whether $\lim_{x \to a} f(x)$ exists based on the existence of these two limits.

Proof/Disproof:

To determine if this is true, we need to investigate whether knowing $\lim_{x \to a} [f(x) \cdot g(x)]$ and $\lim_{x \to a} g(x)$ guarantees the existence of $\lim_{x \to a} f(x)$.

A counterexample disproves this:

• Let $f(x) = \frac{1}{x - a}$ and $g(x) = x - a$. Notice: $f(x) \cdot g(x) = 1 \quad \text{(constant function)}.$ Hence, $\lim_{x \to a} [f(x) \cdot g(x)] = 1$.
• Meanwhile, $\lim_{x \to a} g(x) = 0$.
• However, $\lim_{x \to a} f(x) = \lim_{x \to a} \frac{1}{x - a}$ does not exist because the function blows up to infinity as $x \to a$.

Thus, the statement is false. The existence of the product limit $\lim_{x \to a} [f(x) \cdot g(x)]$ and $\lim_{x \to a} g(x)$ does not guarantee that $\lim_{x \to a} f(x)$ exists.

Question (b):

Statement: IF $\lim_{x \to a} f(x)$ does not exist and $\lim_{x \to a} g(x)$ does not exist, THEN $\lim_{x \to a} [f(x) + g(x)]$ does not exist.

Proof/Disproof:

We need to analyze whether the non-existence of both individual limits implies that the sum of the functions does not have a limit.

This is false, as we can construct a counterexample:

• Let $f(x) = -\frac{1}{x - a}$ and $g(x) = \frac{1}{x - a}$.
• Both $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ do not exist because they go to $-\infty$ and $+\infty$, respectively.
• However, $f(x) + g(x) = 0$, which is constant.
• Hence, $\lim_{x \to a} [f(x) + g(x)] = 0$, which exists.

Thus, the statement is false. Even if both $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ do not exist, their sum can still have a finite limit.

Let me know if you'd like more detailed steps on these proofs or further clarifications!

Follow-up Questions:

1. How does the epsilon-delta definition relate to product or sum limits in general?
2. What are other possible counterexamples where limits behave unexpectedly?
3. How can we use the epsilon-delta definition to prove the existence of a limit rigorously?
4. How can limits that approach infinity or fail to exist affect the behavior of related functions?
5. Are there cases where the sum of non-existent limits might still fail to exist, despite these counterexamples?

Tip:

When tackling limit proofs, always look for counterexamples before assuming a statement is true. These are often the simplest way to disprove general claims.