https://file.aiquickdraw.com/mathbot/file/1727658746082ihmfmsgc.jpg

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.

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### **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.

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### **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 \( t = 20 \).

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### **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 \( x \).

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### **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.

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### **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 \( x \in [-3,5] \).

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### **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.

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### **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.

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### **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.

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### **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 \( x \to 2 \).

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

Use the following piecewise functions, graphs, and tables to answer parts (a) through (i). The problems involve evaluating limits, functions, and specific values based on graphs and piecewise definitions.

Explore advanced calculus concepts with this problem set involving limits, piecewise functions, and graphs. Evaluate limits at specific points, use function compositions, and interpret derivatives from graphical data. Suitable for Grades 11-12 and early college students.

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