How do I us the Limit definition of derivative on #f(x)=tan(x)#?
By Limit Definition,
by taking the common denominator,
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To use the limit definition of the derivative on ( f(x) = \tan(x) ), follow these steps:

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

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

Use the tangent addition formula: [ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1  \tan(a)\tan(b)} ]

Apply the tangent addition formula to the numerator: [ \tan(x + h) = \frac{\tan(x) + \tan(h)}{1  \tan(x)\tan(h)} ]

Substitute this expression into the derivative definition: [ f'(x) = \lim_{h \to 0} \frac{\frac{\tan(x) + \tan(h)}{1  \tan(x)\tan(h)}  \tan(x)}{h} ]

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

Combine like terms and factor out ( \tan(x) ): [ f'(x) = \lim_{h \to 0} \frac{\tan(h)}{h(1  \tan(x)\tan(h))} ]

As ( h ) approaches 0, ( \tan(h) ) approaches ( 0 ): [ f'(x) = \frac{0}{1  \tan(x)(0)} = 0 ]
Therefore, the derivative of ( f(x) = \tan(x) ) is ( f'(x) = 0 ).
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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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