<h2>
Part 1.</h2>
Answer:
The limit of the difference quotient is simply the derivative, so we can express this as follows:
So our function is:
Taking the derivative, we have:
So the correct option is:
<h2>Part 2.</h2>
Answer:
The equation of the line that passes through the same point can be found as:
Where , so we need to find . Plug in that x-value in the function we have:
And the slope is:
Then, the equation of the line is:
<h2 /><h2>Part 3.</h2>
Answer: Shown below
As you can see below, the graph of the function of is continuous. This is so because we have plotted a polynomial function whose domain is the set of all real numbers. So the function is defined at the point , so the derivative exists at this point, hence we can calculate a tangent line there. In conclusion, we get the graph shown below. The blue line is the tangent line while the red curve is the graph of