Using the limit definition, how do you differentiate #f(x) =sqrt(x−3)#?

Answer 1

See the explanation section below.

The crucial step is to use the following to remove the square roots from the numerator.

#(sqrt(x+h-3)-sqrt(x-3))/h = ((sqrt(x+h-3)-sqrt(x-3)))/h ((sqrt(x+h-3)+sqrt(x-3)))/((sqrt(x+h-3)+sqrt(x-3)))#
# = (x+h-3-(x-3))/(h(sqrt(x+h-3)+sqrt(x-3))#
# = 1/(sqrt(x+h-3)+sqrt(x-3))#
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Answer 2

Using the limit definition of the derivative, the derivative of f(x) = sqrt(x - 3) is:

f'(x) = lim[h->0] [(sqrt(x + h - 3) - sqrt(x - 3)) / h]

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Answer 3

To differentiate ( f(x) = \sqrt{x - 3} ) using the limit definition of the derivative, we start by finding the difference quotient:

[ \frac{f(x + h) - f(x)}{h} ]

Substitute ( f(x) = \sqrt{x - 3} ) into the difference quotient:

[ \frac{\sqrt{x + h - 3} - \sqrt{x - 3}}{h} ]

To simplify this expression, we multiply both the numerator and denominator by the conjugate of the numerator:

[ \frac{(\sqrt{x + h - 3} - \sqrt{x - 3})(\sqrt{x + h - 3} + \sqrt{x - 3})}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]

Expanding the numerator and canceling out common terms yields:

[ \frac{(x + h - 3) - (x - 3)}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]

Simplifying further gives:

[ \frac{h}{h(\sqrt{x + h - 3} + \sqrt{x - 3})} ]

Now, as ( h ) approaches 0, the expression becomes:

[ \lim_{h \to 0} \frac{1}{\sqrt{x + h - 3} + \sqrt{x - 3}} ]

Now, evaluate the limit as ( h ) approaches 0:

[ \lim_{h \to 0} \frac{1}{\sqrt{x + h - 3} + \sqrt{x - 3}} = \frac{1}{2\sqrt{x - 3}} ]

Therefore, the derivative of ( f(x) = \sqrt{x - 3} ) using the limit definition is:

[ f'(x) = \frac{1}{2\sqrt{x - 3}} ]

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Answer from HIX Tutor

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.

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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