How to use the alternate definition to find the derivative of #f(x)=sqrt(x+3)# at x=1?
Here are a couple ways you can do the limit calculation for the derivative. Both methods involve "rationalizing the numerator" (not the denominator) as a trick to help you calculate the limits.
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To find the derivative of ( f(x) = \sqrt{x + 3} ) at ( x = 1 ) using the alternate definition, follow these steps:

Determine the alternate definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ]

Substitute ( x = 1 ) into the function: [ f(1) = \sqrt{1 + 3} = \sqrt{4} = 2 ]

Substitute ( x = 1 ) and ( h = 0 ) into the alternate definition: [ f'(1) = \lim_{h \to 0} \frac{f(1 + h)  f(1)}{h} ]

Substitute ( f(1) = 2 ) into the expression: [ f'(1) = \lim_{h \to 0} \frac{\sqrt{1 + h + 3}  2}{h} ]

Simplify the expression: [ f'(1) = \lim_{h \to 0} \frac{\sqrt{h + 4}  2}{h} ]

Rationalize the numerator: [ f'(1) = \lim_{h \to 0} \frac{\sqrt{h + 4}  2}{h} \times \frac{\sqrt{h + 4} + 2}{\sqrt{h + 4} + 2} ] [ f'(1) = \lim_{h \to 0} \frac{h}{h(\sqrt{h + 4} + 2)} ]

Cancel out ( h ): [ f'(1) = \lim_{h \to 0} \frac{1}{\sqrt{h + 4} + 2} ]

Evaluate the limit as ( h ) approaches ( 0 ): [ f'(1) = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4} ]
Therefore, the derivative of ( f(x) = \sqrt{x + 3} ) at ( x = 1 ) using the alternate definition is ( \frac{1}{4} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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