Math Problem Statement

Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false.

Solution

Let’s go through each statement one by one and analyze whether each is true or false.

  1. If limx5f(x)=6\lim_{x \to 5} f(x) = 6 and limx5g(x)=0\lim_{x \to 5} g(x) = 0, then limx5(f(x)g(x))\lim_{x \to 5} \left( \frac{f(x)}{g(x)} \right) does not exist.

    • When the limit of the denominator g(x)g(x) is 0 while the limit of the numerator f(x)f(x) is non-zero (here, 6), the limit limx5(f(x)g(x))\lim_{x \to 5} \left( \frac{f(x)}{g(x)} \right) typically does not exist due to division by zero.
    • Answer: True
  2. If f(6)f'(6) exists, then the limit limx6f(x)\lim_{x \to 6} f(x) is f(6)f(6).

    • The existence of f(6)f'(6) implies that f(x)f(x) is continuous at x=6x = 6, meaning limx6f(x)=f(6)\lim_{x \to 6} f(x) = f(6).
    • Answer: True
  3. If f(x)f(x) is differentiable at aa, then f(x)f(x) is continuous at aa.

    • Differentiability at a point implies continuity at that point. If f(x)f(x) is differentiable at aa, then f(x)f(x) must also be continuous at aa.
    • Answer: True
  4. If p(x)p(x) is a polynomial, then limx6p(x)\lim_{x \to 6} p(x) is p(6)p(6).

    • Polynomials are continuous everywhere. Therefore, limx6p(x)=p(6)\lim_{x \to 6} p(x) = p(6).
    • Answer: True
  5. If f(x)f(x) is continuous at aa, then f(x)f(x) is differentiable at aa.

    • Continuity at a point does not guarantee differentiability. A function can be continuous but not differentiable (e.g., f(x)=xf(x) = |x| at x=0x = 0).
    • Answer: False

Summary of Answers:

  1. True
  2. True
  3. True
  4. True
  5. False

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

Related Questions to Expand Understanding:

  1. What are the conditions required for differentiability at a point?
  2. Can you provide an example of a function that is continuous but not differentiable at a certain point?
  3. How does the Squeeze Theorem relate to limits involving division by zero?
  4. Why does differentiability imply continuity, but not vice versa?
  5. What are some properties of polynomial functions regarding limits and continuity?

Tip:

When evaluating limits involving fractions, always check if the limit of the denominator is zero, as this can indicate a potential for non-existence or the need for further analysis, such as applying L'Hôpital's Rule.

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Differentiability
Polynomials

Formulas

\lim_{x \to a} f(x) = L
f'(a)
f(x) is continuous at a
f(x) is differentiable at a

Theorems

Limit theorem
Continuity and differentiability
Polynomial continuity

Suitable Grade Level

Undergraduate Calculus