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Let's analyze and solve the problems shown in the image. The problems focus on finding limits and determining functions based on given limit conditions.

### Problem 49:
\[
\text{If } \lim_{x \to 4} \frac{f(x) - 5}{x - 2} = 1, \text{ find } \lim_{x \to 4} f(x).
\]

#### Solution:
We can approach this problem by recognizing that the limit provided resembles the definition of a derivative. Specifically, the limit can be interpreted as the derivative of \( f(x) \) at some point. However, directly, we can rewrite the expression as:

\[
\lim_{x \to 4} \frac{f(x) - 5}{x - 2} = 1
\]

This suggests that \( f(x) \) behaves like a linear function near \( x = 4 \), and can be written as:

\[
f(x) \approx m(x - 2) + 5
\]

Given that the limit equals 1:

\[
m = 1
\]

So, \( f(x) \approx 1 \cdot (x - 2) + 5 = x + 3 \). Therefore:

\[
\lim_{x \to 4} f(x) = 4 + 3 = 7
\]

### Problem 50:
\[
\text{If } \lim_{x \to 0} \frac{f(x)}{x^2} = 1, \text{ find } (a) \lim_{x \to 0} f(x) \text{ and } (b) \lim_{x \to 0} \frac{f(x)}{x}.
\]

#### Solution:
- (a) We have:

\[
\lim_{x \to 0} \frac{f(x)}{x^2} = 1
\]

This implies:

\[
f(x) \approx x^2 \text{ as } x \to 0
\]

Thus:

\[
\lim_{x \to 0} f(x) = 0
\]

- (b) We need to find:

\[
\lim_{x \to 0} \frac{f(x)}{x}
\]

Given \( f(x) \approx x^2 \), we have:

\[
\lim_{x \to 0} \frac{x^2}{x} = \lim_{x \to 0} x = 0
\]

### Problem 51:
- **(a)** \(\text{If } \lim_{x \to 3} \frac{f(x) - 5}{x - 2} = 3, \text{ find } \lim_{x \to 2} f(x).\)

- **(b)** \(\text{If } \lim_{x \to 2} \frac{f(x)}{x - 2} = 4, \text{ find } \lim_{x \to 2} f(x).\)

#### Solution for 51 (a):
Similar to Problem 49, the limit resembles a derivative form, implying:

\[
\lim_{x \to 2} f(x) = 5
\]

#### Solution for 51 (b):
Since \( \lim_{x \to 2} \frac{f(x)}{x - 2} = 4 \):

\[
f(x) \approx 4(x - 2) \text{ near } x = 2
\]

So:

\[
\lim_{x \to 2} f(x) = 0
\]

### Problem 52:
\[
\text{If } \lim_{x \to 0} \frac{f(x)}{x^2} = 1, \text{ find } (a) \lim_{x \to 0} f(x) \text{ and } (b) \lim_{x \to 0} \frac{f(x)}{x}.
\]

#### Solution:
This is the same as Problem 50, so:

- (a) \(\lim_{x \to 0} f(x) = 0\)
- (b) \(\lim_{x \to 0} \frac{f(x)}{x} = 0\)

Would you like more detailed explanations on any specific problem? Here are five related questions that might help expand your understanding:

1. How do you identify when a limit expression resembles the definition of a derivative?
2. What techniques can you use when limits involve indeterminate forms?
3. How can you find the exact function \( f(x) \) given a limit condition?
4. Why does the behavior of \( f(x) \) near the point of interest influence the limit?
5. How do higher-order limits (like involving \( x^2 \) or higher) affect the solution?

**Tip:** Always check if the limit expression can be interpreted as the definition of a derivative, as this often simplifies solving the problem.

The query submitted by the user contains a set of mathematical problems focused on finding limits and determining functions based on given limit conditions.

Explore limit problems that involve concepts such as derivatives and linear approximations. This detailed solution helps understand how to find limits and approximate functions near given points, suitable for Grades 11-12 students.

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