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How do you find f'(2) using the limit definition given #f(x)= 9-x^2#?

Answer 1

Charles Anctil

-4

Using the limit definition, you get:

#f'(2)=lim_(x->2)(f(x)-f(2))/(x-2)#

#=lim_(x->2)((9-x^2)-(9-2^2))/(x-2)#

#=lim_(x->2)(9-x^2-5)/(x-2)#

#=lim_(x->2)(4-x^2)/(x-2)#

#=lim_(x->2)-(x^2-4)/(x-2)#

#=lim_(x->2)-(cancel((x-2))(x+2))/cancel(x-2)#

#=-(2+2)=-4#

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Answer 2

Paisley Johnson

-4

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Answer 3

Scarlett Allison

To find ( f'(2) ) using the limit definition given ( f(x) = 9 - x^2 ), follow these steps:

Start with the function ( f(x) = 9 - x^2 ).
Find ( f'(x) ), the derivative of ( f(x) ), using the limit definition of the derivative.
Evaluate ( f'(x) ) at ( x = 2 ).

Here's how to proceed:

( f(x) = 9 - x^2 )
( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} )
( f'(x) = \lim_{h \to 0} \frac{(9 - (x + h)^2) - (9 - x^2)}{h} ) 4

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Answer from HIX Tutor

When evaluating a one-sided 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.