(1 point) Let f(x)=(9)/(4pi -8x) Then the slope of the tangent : line to the graph of y=f(x) at the point (pi ,-(9)/(4pi )) is the limit as x tends to 3.14 of the following expression square The value of this limit is square It follows that the equation of the tangent line is y= square

To find the slope of the tangent line to the graph of $y=f(x)$ at the point $(\pi, -\frac{9}{4\pi})$, we need to find the derivative of $f(x)$ and evaluate it at $x=\pi$.<br /><br />The derivative of $f(x)$ is given by:<br />$$f'(x) = \frac{d}{dx}\left(\frac{9}{4\pi - 8x}\right)$$<br /><br />Using the quotient rule, we have:<br />$$f'(x) = \frac{9 \cdot (-8)}{(4\pi - 8x)^2} = \frac{-72}{(4\pi - 8x)^2}$$<br /><br />Now, we can evaluate $f'(x)$ at $x=\pi$:<br />$$f'(\pi) = \frac{-72}{(4\pi - 8\pi)^2} = \frac{-72}{(-4\pi)^2} = \frac{-72}{16\pi^2} = \frac{-9}{4\pi^2}$$<br /><br />Therefore, the slope of the tangent line to the graph of $y=f(x)$ at the point $(\pi, -\frac{9}{4\pi})$ is $\frac{-9}{4\pi^2}$.<br /><br />Next, we need to find the value of the limit as $x$ tends to $\pi$ of the expression $\frac{f(x) - f(\pi)}{x - \pi}$.<br /><br />Substituting $f(x) = \frac{9}{4\pi - 8x}$ and $f(\pi) = -\frac{9}{4\pi}$, we have:<br />$$\lim_{x \to \pi} \frac{f(x) - f(\pi)}{x - \pi} = \lim_{x \to \pi} \frac{\frac{9}{4\pi - 8x} + \frac{9}{4\pi}}{x - \pi}$$<br /><br />Simplifying the expression, we get:<br />$$\lim_{x \to \pi} \frac{9}{(4\pi - 8x)(4\pi)} \cdot \frac{4\pi}{x - \pi} = \lim_{x \to \pi} \frac{9}{(4\pi - 8x)(4\pi)(x - \pi)}$$<br /><br />As $x$ approaches $\pi$, the denominator approaches zero, so the limit is undefined.<br /><br />Therefore, the value of the limit is undefined.<br /><br />Finally, we can use the point-slope form of a linear equation to find the equation of the tangent line:<br />$$y - y_1 = m(x - x_1)$$<br /><br />Substituting $m = \frac{-9}{4\pi^2}$, $x_1 = \pi$, and $y_1 = -\frac{9}{4\pi}$, we have:<br />$$y + \frac{9}{4\pi} = \frac{-9}{4\pi^2}(x - \pi)$$<br /><br />Simplifying the equation, we get:<br />$$y = \frac{-9}{4\pi^2}(x - \pi) - \frac{9}{4\pi}$$<br /><br />Therefore, the equation of the tangent line is $y = \frac{-9}{4\pi^2}(x - \pi) - \frac{9}{4\pi}$.