SageMath has the ability to compute the limits of certain functions. For example, to compute \( \lim\limits_{x \to 1} \dfrac{x^2-2x+1}{x-1} \) you would type:

f(x)=(x^2-2*x+1)/(x-1)
limit(f(x), x=1)

Try this in the SageMath cell below.  

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3a. We can use the limit command in SageMath to verify our answer to Question 2b. Recall that the slope of the secant line of \( f(x) \) from \( x = a \) to \( x = b \) is given by \[ m_{\mathrm{sec}} = \frac{f(b) – f(a)}{b – a} \] If we fix a particular value of \( a \), say \( a = 1 \), this becomes a function of \( b \). In this case, we can define the function m(b)=(f(b)-f(1))/(b-1) and take the limit as \( b \) approaches \( a \). Use the SageMath cell below to compute this limit for the function \( f(x) \) assigned in your lab report. Don’t forget to define \(f(x)\) before defining \( m(b) \).

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3b. The computation you did in the previous question was computing the derivative of \( f(x) \) at \( x = 1 \) using the limit definition of the derivative. SageMath can compute derivatives directly. For example, to compute \( f'(3) \), we define \( f(x) \), then write

df(x)=diff(f,x)
df(3)

Use the SageMath cell below to compute \( f'(1) \) for the function \( f(x) \) assigned in your lab report.

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3c. Find the equation, in point-slope form, of the line passing through \( (1, f(1)) \) with slope equal to the result from Question 3b. Record your answer in your lab report.

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3d. Define a function \( g(x) \) to be the line from Question 3c (that is, solve for \( y \), then replace \( y \) with \( g(x) \)). Plot \( f(x) \) and \( g(x) \) together on the interval \( 0 \le x \le 5 \).

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