The Limit Represents The Derivative Of Some Function F At Some Number A. State Such An F And A.
The limit represents the derivative of some function f(x) at some number a. Find f and a.
lim (h->0) of ((7+h)^2-49)/h
Suppose that the function is `f(x)=x^2` and `a = 7` .
You need to find the derivative of the function at x=a=7, hence, using the limit definition of derivatives yields:
`f'(x) = lim_(h-gt0) (f(x+h)-f(x))/h`
`f'(x) = lim_(h-gt0) ((x+h)^2-x^2)/h`
Expanding the binomial yields:
`f'(x) = lim_(h-gt0) (x^2 + 2xh + h^2 - x^2)/h`
Reducing like terms yields:
`f'(x) = lim_(h-gt0) (2xh + h^2)/h`
Factoring out h yields:
`f'(x) = lim_(h-gt0) h(2x + h)/h =gt f'(x) = lim_(h-gt0) (2x + h) = 2x`
Equating `((x+h)^2-x^2)/h` and `((7+h)^2-49)/h` yields x + h = 7 + h and `x^2 = 49` .
Notice that het relations gives `x_(1,2) = +-7` but the first relation x + h = 7 + h excludes the value -7, hence x = 7.
Hence, evaluating the function yields that `f(x) = x^2` and a=7 => f'(7) = 14.