a. Use the definition $m_{\tan }=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$ to find the slope of the line tangent to the graph of $f$ at $P$ b. Determine an equation of the tangent line at $P$. \[ f(x)=x^{2}+4, P(5,29) \]

Step 1 ：Given the function \(f(x) = x^2 + 4\) and the point \(P(5,29)\).

Step 2 ：The slope of the tangent line to the graph of the function at a point is given by the derivative of the function at that point.

Step 3 ：Find the derivative of the function \(f(x) = x^2 + 4\), which is \(f'(x) = 2x\).

Step 4 ：Evaluate the derivative at \(x = 5\) to find the slope of the tangent line, which is \(f'(5) = 2*5 = 10\).

Step 5 ：Use the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), to find the equation of the tangent line. Here, \(m = 10\) and \((x_1, y_1) = (5, 29)\).

Step 6 ：Substitute the values into the equation to get \(y - 29 = 10(x - 5)\).

Step 7 ：Simplify the equation to get \(y = 10x - 21\).

Step 8 ：Final Answer: The equation of the tangent line to the graph of the function \(f(x) = x^2 + 4\) at the point \(P(5,29)\) is \(\boxed{y = 10x - 21}\).