How do you use the limit definition to find the slope of the tangent line to the graph #f(x)= x^2-2x+4# at (0,4)?

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Answer 1 · 2017-05-07T18:18:55

#m=-2#

The limit definition of a derivative is given by:

#f'(x)=lim_(h->0) (f(x+h)-f(x))/h#

Here, the derivative is:

#f'(x)=lim_(h->0)((x+h)^2-2(x+h)+4-x^2+2x-4)/h#

#=lim_(h->0)(x^2+2hx+h^2-2x-2h-x^2+2x)/h#

#=lim_(h->0)(2hx)/h-(2h)/h+h^2/h#

#=2x-2+lim_(h->0)h#

#=2x-2#

#f'(x)=2x-2#

We have now found the gradient of the tangent at any point. All that's left is to evaluate it at #x=0#.

#f'(0)=2(0)-2=-2#