Question

a. use the definition $m_{tan}=lim_{h \to 0}\frac{f(a + h)-f(a)}{h}$ 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
$f(x)=\frac{1}{2 + 3x},p(1,\frac{1}{5})$
a. when using the definition $m_{tan}=lim_{h \to 0}\frac{f(a + h)-f(a)}{h}$, which is an expression for the limit to use to find the slope of the tangent line at p?
oa. $lim_{h \to 0}\frac{f(1 + h)-\frac{1}{5}}{1 + h}$
ob. $lim_{h \to 0}\frac{f(1 + h)-\frac{1}{5}}{h}$
oc. $lim_{h \to 0}\frac{f(1 + h)+\frac{1}{5}}{h}$

Explanation:

Step1: Identify \(a\) from the point \(P\)

The point \(P=(1,\frac{1}{5})\), so \(a = 1\) and \(f(a)=f(1)=\frac{1}{2 + 3\times1}=\frac{1}{5}\).

Step2: Substitute into the slope - formula

The formula for the slope of the tangent line is \(m_{\tan}=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}\). Substituting \(a = 1\) gives \(m_{\tan}=\lim_{h
ightarrow0}\frac{f(1 + h)-f(1)}{h}=\lim_{h
ightarrow0}\frac{f(1 + h)-\frac{1}{5}}{h}\).

Answer:

B. \(\lim_{h
ightarrow0}\frac{f(1 + h)-\frac{1}{5}}{h}\)