# Use the limit definition to find the derivative of f(x)=3/sqrtx at the number a, and use it to find the equation of the tangent line to the curve at the point (1,3)?

##
f(x)= #3/sqrtx#

f(x)=

# f'(a) = (-3)/(a(2sqrt(a)) )#

# y = -3/2 x + 9/2 #

Using the limit definition of the derivative, we have:

And we get the equation:

Which we can verify graphically:

graph{(y-3/sqrt(x))(y +3/2 x - 9/2)=0 [-5, 10, -5, 10]}

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To find the derivative of (f(x) = \frac{3}{\sqrt{x}}) using the limit definition, we'll use the formula:

[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} ]

First, let's find (f(a + h)):

[ f(a + h) = \frac{3}{\sqrt{a + h}} ]

Next, let's find (f(a)):

[ f(a) = \frac{3}{\sqrt{a}} ]

Now, let's plug these into the formula:

[ f'(a) = \lim_{h \to 0} \frac{\frac{3}{\sqrt{a + h}} - \frac{3}{\sqrt{a}}}{h} ]

To simplify this expression, we'll rationalize the numerator:

[ f'(a) = \lim_{h \to 0} \frac{3\sqrt{a} - 3\sqrt{a + h}}{h\sqrt{a}\sqrt{a + h}} ]

[ f'(a) = \lim_{h \to 0} \frac{3\sqrt{a} - 3\sqrt{a + h}}{h\sqrt{a}\sqrt{a + h}} \cdot \frac{\sqrt{a} + \sqrt{a + h}}{\sqrt{a} + \sqrt{a + h}} ]

[ f'(a) = \lim_{h \to 0} \frac{3a - 3(a + h)}{h\sqrt{a}(a + h)} ]

[ f'(a) = \lim_{h \to 0} \frac{-3h}{h\sqrt{a}(a + h)} ]

[ f'(a) = \lim_{h \to 0} \frac{-3}{\sqrt{a}(a + h)} ]

[ f'(a) = \frac{-3}{\sqrt{a} \cdot a} ]

Now, to find the equation of the tangent line at the point (1, 3), we need to find the slope of the tangent line at (x = 1).

[ f'(1) = \frac{-3}{\sqrt{1} \cdot 1} = -3 ]

So, the slope of the tangent line is (m = -3).

Using the point-slope form of the equation of a line, (y - y_1 = m(x - x_1)), where ((x_1, y_1)) is the point (1, 3), we plug in the values:

[ y - 3 = -3(x - 1) ]

Now, we can simplify this equation to get the equation of the tangent line.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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