Question

math 171 calculus projects 1, can you please help with the math questions 1,2, and 3 they have been attached below.

MATH 171 - CALCULUS I

PROJECT l

Note: Make sure to show all supporting work to receive full credit. Your answers should be stated in the context of the problem and include appropriate units where applicable.

1. Obtain graphical and numerical evidence concerning the existence of \(\lim _{x \rightarrow 0} \frac{50 x^{2}}{\sin x+50 x^{2}} .\) You must provide:

- A graph using a window that shows the behavior of the function very close to 0. Your graph must be properly labeled. Write an observation about the limit based on your graph.

- Two tables showing the behavior of the function very close to zero on either side. Your tables must be properly labeled. Write an observation about the limit based on each table.

- A final conclusion about the existence of the limit. If the limit does exist, clearly state the value of the limit using correct limit notation. If the limit does not exist, say why not.

2. a. Write the precise mathematical definition of continuity.

b. Explain in your own words what it means for a function to be continuous at a point. What will be some characteristics of the graph of the function?

c. Suppose the function \(h(x)\) is defined as follows: 

- Use the definition of continuity to find values of \(a\) and \(b\) that make \(h(x)\) continuous at \(\quad x=2\)

- Show each step of the definition of continuity, including both left and right limits.

- Clearly explain how you arrived at your answer and obtain a printout of the graph of your function.

d. Using your answer to part (c), is \(h(x)\) continuous everywhere? How do you know?

3. Sketch a graph of one possible function \(f(x)\) for which all of the following conditions are true. Write down what each condition tells you about the graph of \(f(x)\)

a. \(\quad \lim _{x \rightarrow 3^{-}} f(x)=\infty\)

b. \(\quad \lim _{x \rightarrow 3^{+}} f(x)=-\infty\)

c. \(f(0)=1\)

d. \(f(3)=4\)

e. \(\lim _{x \rightarrow-\infty} f(x)=-\infty\)

f. \(\lim _{x \rightarrow \infty} f(x)=1\)

Answer 1