Let's evaluate the limit and the given statements carefully.

We are given:

$\lim_{x \to \infty} \frac{x + \sin x}{x}$

and two functions: $f(x) = x + \sin x$ and $g(x) = x$.

Statement a. $g'(x) \neq 0$ on $(M, \infty)$ for some $M > 0$

• $g(x) = x$, so $g'(x) = 1$.
• Since $g'(x) = 1$ is non-zero for all $x \in (M, \infty)$, this statement is True.

Statement b. $f$ and $g$ are differentiable on $(M, \infty)$ for some $M > 0$

• $f(x) = x + \sin x$ is differentiable because both $x$ and $\sin x$ are differentiable.
• $g(x) = x$ is also differentiable since it's a simple linear function.
• Thus, both functions are differentiable on $(M, \infty)$, so this statement is True.

Statement c. $\lim_{x \to \infty} \frac{f(x)}{g(x)}$ does not exist

Let's compute the limit:

$\lim_{x \to \infty} \frac{x + \sin x}{x} = \lim_{x \to \infty} \left( 1 + \frac{\sin x}{x} \right)$

Since $\frac{\sin x}{x} \to 0$ as $x \to \infty$, we get:

$\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1$

This limit exists, so this statement is False.

Statement d. $f$ and $g$ are continuous on $(M, \infty)$ for some $M > 0$

• $f(x) = x + \sin x$ is continuous because both $x$ and $\sin x$ are continuous functions.
• $g(x) = x$ is continuous as a linear function.
• Thus, both $f$ and $g$ are continuous on $(M, \infty)$, making this statement True.

Statement e. $\lim_{x \to \infty} \frac{f'(x)}{g'(x)}$ does not exist

Let's differentiate $f(x)$ and $g(x)$:

• $f'(x) = 1 + \cos x$
• $g'(x) = 1$

Now, we compute:

$\frac{f'(x)}{g'(x)} = 1 + \cos x$

Since $\cos x$ oscillates between $-1$ and $1$, the expression $1 + \cos x$ oscillates between $0$ and $2$. Therefore, the limit does not exist because of this oscillation. Thus, this statement is True.

Statement f. $\lim_{x \to \infty} \frac{f'(x)}{g'(x)}$ exists

From the reasoning above, the limit does not exist due to the oscillating behavior of $\cos x$, so this statement is False.

Statement g. $\lim_{x \to \infty} \frac{f(x)}{g(x)}$ is a $\frac{\infty}{\infty}$ form

• When we initially compute the limit, the form $\frac{f(x)}{g(x)} = \frac{x + \sin x}{x}$ is indeed an indeterminate form $\frac{\infty}{\infty}$ because both the numerator and denominator go to infinity as $x \to \infty$.
• Therefore, this statement is True.

Summary of True/False:

• a. True
• b. True
• c. False
• d. True
• e. True
• f. False
• g. True

Would you like more details on any of these steps or explanations?

Here are 5 related questions:

1. How do you apply L'Hôpital's Rule to limits involving $\frac{\infty}{\infty}$ forms?
2. What is the behavior of trigonometric functions like $\sin x$ and $\cos x$ as $x \to \infty$?
3. How does the differentiability of functions affect their limits?
4. What are the conditions for the existence of limits involving oscillating functions?
5. What does the term "indeterminate form" mean in calculus?

Tip: When limits are in indeterminate forms like $\frac{\infty}{\infty}$, consider applying L'Hôpital's Rule to simplify the expression.