How do you find f'(5) using the limit definition given #f(x) = −2/(x + 1)#?
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To find ( f'(5) ) using the limit definition given ( f(x) = \frac{2}{x + 1} ), follow these steps:

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

Substitute ( a = 5 ) into the definition: [ f'(5) = \lim_{{h \to 0}} \frac{{f(5 + h)  f(5)}}{h} ]

Substitute the function ( f(x) ): [ f'(5) = \lim_{{h \to 0}} \frac{{\frac{2}{(5 + h) + 1}  \frac{2}{5 + 1}}}{h} ]

Simplify the expression: [ f'(5) = \lim_{{h \to 0}} \frac{{\frac{2}{4 + h} + \frac{2}{4}}}{h} ]

Combine fractions and simplify further: [ f'(5) = \lim_{{h \to 0}} \frac{{2(4)  2(4 + h)}}{h(4)(4 + h)} ] [ f'(5) = \lim_{{h \to 0}} \frac{{8  2(4 + h)}}{h(4)(4 + h)} ]

Expand and simplify the expression: [ f'(5) = \lim_{{h \to 0}} \frac{{8 + 2h}}{h(4)(4 + h)} ] [ f'(5) = \lim_{{h \to 0}} \frac{{8 + 2h}}{4h(4 + h)} ] [ f'(5) = \lim_{{h \to 0}} \frac{{8 + 2h}}{4h^2 + 16h} ]

Evaluate the limit as ( h ) approaches 0: [ f'(5) = \frac{8}{4(0)^2 + 16(0)} ] [ f'(5) = \frac{8}{0} ]
Since the denominator approaches zero as ( h ) approaches 0, we cannot evaluate the derivative at ( x = 5 ) using the limit definition. Thus, the derivative ( f'(5) ) is undefined.
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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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