Let's analyze each statement and determine whether it's true (T) or false (F).

1. Statement: "If $p(x)$ is a polynomial, then the limit as $x \to 4$ of $p(x)$ is $p(4)$."

   • Explanation: For any polynomial $p(x)$, the limit as $x$ approaches a specific point $a$ is simply the value of $p(x)$ at that point, $p(a)$. This follows from the fact that polynomials are continuous functions.
   • Answer: True (T)

2. Statement: $\lim_{x \to 3} \frac{x^2 + 4x - 20}{x^2 + 5x - 22} = \lim_{x \to 3} \frac{x^2 + 4x - 20}{x^2 + 5x - 22}$

   • Explanation: First, let's simplify $\frac{x^2 + 4x - 20}{x^2 + 5x - 22}$ as $x \to 3$.

     • Factor $x^2 + 4x - 20 = (x + 6)(x - 2)$.
     • Factor $x^2 + 5x - 22 = (x + 6)(x - 3)$.

     The expression simplifies to $\frac{(x + 6)(x - 2)}{(x + 6)(x - 3)} = \frac{x - 2}{x - 3}$. As $x \to 3$, this simplifies to a form that does not exist (it leads to division by zero).

   • Answer: False (F)

3. Statement: If $\lim_{x \to 3} f(x) = 4$ and $\lim_{x \to 3} g(x) = 0$, then $\lim_{x \to 3} \frac{f(x)}{g(x)}$ does not exist.

   • Explanation: Since $g(x)$ approaches zero and $f(x)$ approaches a non-zero value (4), the limit $\frac{f(x)}{g(x)}$ will tend towards infinity or negative infinity, which means the limit does not exist in the finite sense.
   • Answer: True (T)

4. Statement: If $f(x)$ is differentiable at $a$, then $f(x)$ is continuous at $a$.

   • Explanation: Differentiability at a point $a$ implies continuity at that point. If $f(x)$ is differentiable at $a$, it must also be continuous at $a$.
   • Answer: True (T)

5. Statement: If $\lim_{x \to 3} f(x) = 0$ and $\lim_{x \to 3} g(x) = 0$, then $\lim_{x \to 3} \frac{f(x)}{g(x)}$ does not exist.

   • Explanation: This is a case of an indeterminate form $\frac{0}{0}$. The existence of the limit depends on the behavior of $f(x)$ and $g(x)$ as $x$ approaches 3. For example, if both functions have the same linear behavior near $x = 3$, the limit might exist. Therefore, this statement is not always true.
   • Answer: False (F)

Summary of Answers:

1. True (T)
2. False (F)
3. True (T)
4. True (T)
5. False (F)

Would you like more details on any of these answers, or do you have questions?

Related Questions:

1. What does it mean for a function to be continuous at a point?
2. How does differentiability imply continuity?
3. What are indeterminate forms, and how are they resolved?
4. How can limits involving polynomials be simplified?
5. What are some examples where $\frac{f(x)}{g(x)}$ does exist even if $f(x) \to 0$ and $g(x) \to 0$?

Tip:

In limit problems involving rational functions, factoring and canceling common terms often help simplify the expression and resolve indeterminate forms.