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

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

Substitute ( g(x) = \frac{2}{x + 1} ) into the formula: [ g'(x) = \lim_{{h \to 0}} \frac{{\frac{2}{{x + h + 1}}  (\frac{2}{{x + 1}})}}{h} ]

Combine the fractions: [ g'(x) = \lim_{{h \to 0}} \frac{{\frac{2}{{x + h + 1}} + \frac{2}{{x + 1}}}}{h} ]

Find a common denominator: [ g'(x) = \lim_{{h \to 0}} \frac{{\frac{2(x + 1)}{{(x + h + 1)(x + 1)}} + \frac{2(x + h + 1)}{{(x + h + 1)(x + 1)}}}}{h} ]

Combine the fractions: [ g'(x) = \lim_{{h \to 0}} \frac{{\frac{2(x + 1) + 2(x + h + 1)}}{{(x + h + 1)(x + 1)}}}{h} ]

Simplify the numerator: [ g'(x) = \lim_{{h \to 0}} \frac{{\frac{2x  2 + 2x + 2h + 2}}{{(x + h + 1)(x + 1)}}}{h} ]

Simplify further: [ g'(x) = \lim_{{h \to 0}} \frac{{\frac{2h}{{(x + h + 1)(x + 1)}}}}{h} ]

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

Evaluate the limit as ( h ) approaches 0: [ g'(x) = \frac{2}{{(x + 1)^2}} ]
So, the derivative of ( g(x) = \frac{2}{x + 1} ) with respect to ( x ) is ( g'(x) = \frac{2}{{(x + 1)^2}} ).
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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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