How do you use the limit definition of the derivative to find the derivative of #f(x)=sqrt(4x5)#?
By definition of derivative we have:
so:
rationalize the numerator using the identity:
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To use the limit definition of the derivative to find the derivative of f(x) = √(4x  5), follow these steps:

Start with the limit definition of the derivative: f'(x) = lim(h>0) [f(x + h)  f(x)] / h.

Substitute f(x) into the formula: f'(x) = lim(h>0) [√(4(x + h)  5)  √(4x  5)] / h.

Simplify the expression by expanding the square roots: f'(x) = lim(h>0) [(√(4x + 4h  5)  √(4x  5))] / h.

Rationalize the numerator by multiplying by the conjugate: f'(x) = lim(h>0) [(√(4x + 4h  5)  √(4x  5)) * (√(4x + 4h  5) + √(4x  5))] / (h * (√(4x + 4h  5) + √(4x  5))).

Expand and simplify the expression: f'(x) = lim(h>0) [(4x + 4h  5)  (4x  5)] / (h * (√(4x + 4h  5) + √(4x  5))).

Cancel out like terms: f'(x) = lim(h>0) [4h] / (h * (√(4x + 4h  5) + √(4x  5))).

Simplify further: f'(x) = lim(h>0) [4] / (√(4x + 4h  5) + √(4x  5)).

Plug in h = 0: f'(x) = 4 / (√(4x  5) + √(4x  5)).

Simplify the expression: f'(x) = 4 / (2√(4x  5)).

Finally, simplify further: f'(x) = 2 / √(4x  5).
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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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