Explained: "How do you use the limit definition to..."

Abbigail Robles

Answered question

2022-04-22

How do you use the limit definition to find the slope of the tangent line to the graph

f (x) = x^{2} - 2 x + 4

at (0,4)?

Bethany Mills

Beginner2022-04-23Added 14 answers

The limit definition of a derivative is given by:

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

Here, the derivative is:

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

= lim_{h \to 0} \frac{x^{2} + 2 x h + h^{2} - 2 x - 2 h - x^{2} + 2 x}{h}

= lim_{h \to 0} \frac{2 h x}{h} - \frac{2 h}{h} + \frac{h^{2}}{h}

= 2 x - 2 + lim_{h \to 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

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