Let's walk through the problem step by step using the image provided.

First, I will break down each part of the question based on the given piecewise functions, graphs, and tables.

Part (a)

Find a value of $k$ that makes: $\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x).$

Here, we are dealing with the limit of the function $f(x)$ from both the left and right sides as $x \to 3$. The function $f(x)$ is defined as a piecewise function, so we need to evaluate the left-hand limit (for $x \to 3^-$) and the right-hand limit (for $x \to 3^+$) using the appropriate expressions.

• For $x < 3$, use the expression $\frac{4x^2 - 5x - 3}{x + 1}$.
• For $x > 3$, use $kx + 3$.

To find $k$, set the left-hand limit equal to the right-hand limit.

Part (b)

Evaluate $\lim_{t \to 20} R(t)$.

The function $R(t)$ is defined piecewise as:

• For $0 \leq t < 20$, $R(t) = \frac{41}{t/20}$.
• For $t \geq 20$, $R(t) = \frac{1000 + t}{t^2 + 6000}$.

We need to evaluate the limit from both sides (left and right) at $t = 20$.

Part (c)

Evaluate $k(h(p(n(p(x)))))$.

This problem requires us to substitute the value of each function into one another, starting from the innermost function. The function definitions provided include:

• $p(t)$, a piecewise function.
• $h(x)$, which is given by the graph.
• $n(x)$, also provided by a graph.

We’ll compute each of these step by step and evaluate at the necessary $x$.

Part (d)

Compute $\lim_{x \to 4} \frac{m(x)}{3g(x) - n(x)}$.

We need to calculate the limits of each component function:

• $m(x)$, $g(x)$, and $n(x)$ are provided by graphs.
• Evaluate $m(x)$, $g(x)$, and $n(x)$ at $x = 4$, and then use these values to compute the final expression.

Part (e)

Summing the number of times certain values occur for the functions $g(x)$, $h(x)$, $m(x)$, and $n(x)$ on the specified intervals.

We need to count the occurrences:

• $g(x) = 2.5$ on $x \in [0,5]$.
• $h(x) = 1$ on $x \in [0,5]$.
• $m(x) = 0$ on $x \in [-3,5]$.
• $n(x) = -0.5$ on $x \in [-3,5]$.

Part (f)

Evaluate $\frac{3h'(1) - 4h'(2.5)}{h'(3.1) - h'(4.5)}$.

Here we use the derivative of $h(x)$, which is the slope of the graph of $h(x)$, at the specified points $x = 1, 2.5, 3.1, 4.5$. By reading the slopes from the graph, we can compute this expression.

Part (g)

Evaluate $\lim_{x \to \infty} \frac{p(x)}{R(x)}$.

We’ll need to determine the behavior of $p(x)$ and $R(x)$ as $x \to \infty$ by analyzing the asymptotic behavior of both functions based on their piecewise definitions.

Part (h)

Evaluate a complex expression involving multiple functions, using previous results from earlier parts.

This involves substituting results from earlier parts (like $k$ from part (a)) into this nested expression.

Part (i)

Evaluate $\lim_{x \to 2} h(g(x))$.

To solve this, substitute $g(x)$ into $h(x)$ and then find the limit as $x \to 2$.

Let me know which specific part you'd like to focus on, or if you'd like detailed steps for a particular section!