7/17/05 DUE DATE: MATH 171 – CALCULUS IPROJECT 1 – FUNCTIONS, LIMITS, AND CONTINUITYVERSION A1. Find the FunctionNote: For this problem you may NOT present a solution that uses the curve fitting capabilities of Converge, your graphing calculator or any other technology.In each part, provide - A complete explanation of how you arrived at your function - A computer or calculator generated graph of your function showing the designated points - A table showing that the three points belong to your function or come within the specifiedtolerance.a. Consider the points (2, 0), (5, 0), and (8, 0). There is a simple linear function that goes through these points exactly. What is it?b. Find a third degree polynomial (f x ax bx cx d( )    3 2) that goes through the three points in part (a) or comes within 0.1 of doing so. Make sure to write your function in thespecified form in your final answer.c. Find an exponential function of the form f x a b ckx( )   with b 1, 0a , and k 0 that goes through the three points in part (a) or comes within 0.1 of doing so. Make sure to write the final version of your function in the specified form.d. Find a sinusoidal function of the form f x a bx( ) sin( ) with 0a  and 0b  that goes through the three points in part (a) or comes within 0.1 of doing so. e. Find a function that either goes through each of the points (–2, 1), (1, 3), and (4, 0) exactly or comes within 0.1 of doing so. 2. Obtain graphical and numerical evidence concerning the existence of 22050sin 50limxxx x. 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. 3. 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? 4. Suppose the function ( )h x is defined as follows: 23 if 2( ) 3 if 22 if 2x ax xh x xbx x x�- <��= =��+ >��. a. Use the definition of continuity to find values of a and b that make h x( ) continuous at 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 yourfunction.b. Using your answer to part (a), is h x( ) continuous everywhere? How do you know?5. 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. 3lim ( )xf x-�=�b. 3lim ( )xf x+�=-�c. (0) 1f d. (3) 4f e. lim ( )xf x�- �=- �f. lim ( ) 1xf