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Math Help - Using limits to find an equation of a tangent line to the graph of f at given point

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    Using limits to find an equation of a tangent line to the graph of f at given point

    Given:

    \sqrt{x+2} at the point (7,3)

    I need to use the limit definition of :

    m= \lim_{\Delta x\rightarrow 0} \frac{f(x+\Delta x)- f(x)}{\Delta x}

    to find an equation of the tangent line to the graph of f at the given point (above).
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  2. #2
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    Quote Originally Posted by AMaccy View Post
    Given:

    \sqrt{x+2} at the point (7,3)

    I need to use the limit definition of :

    m= \lim_{\Delta x\rightarrow 0} \frac{f(x+\Delta x)- f(x)}{\Delta x}

    to find an equation of the tangent line to the graph of f at the given point (above).
    let h = \Delta x

    f(x) = \sqrt{x+2}

    f(x+h) = \sqrt{x+h+2}

    \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

    \lim{h \to 0} \frac{\sqrt{x+h+2}-\sqrt{x+2}}{h}

    \lim{h \to 0} \frac{\sqrt{x+h+2}-\sqrt{x+2}}{h} \cdot \frac{\sqrt{x+h+2}+\sqrt{x+2}}{\sqrt{x+h+2}+\sqrt{  x+2}}

    \lim{h \to 0} \frac{(x+h+2)-(x+2)}{h(\sqrt{x+h+2}+\sqrt{x+2})}

    \lim_{h \to 0} \frac{1}{\sqrt{x+h+2}+\sqrt{x+2}} = \, ?

    sub in 7 for x as your last step after you find f'(x)

    find the tangent line equation using the point-slope form of a linear equation
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