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ehidna [41]
2 years ago
14

Limit definition for slope of the graph, equation of tangent line point for f(x)=2x^2 at x=(-1)

Mathematics
1 answer:
Tems11 [23]2 years ago
6 0

The slope of the tangent line to f at x=-1 is given by the derivative of f at that point:

f'(-1)=\displaystyle\lim_{x\to-1}\frac{f(x)-f(-1)}{x-(-1)}=\lim_{x\to-1}\frac{2x^2-2}{x+1}

Factorize the numerator:

2x^2-2=2(x^2-1)=2(x-1)(x+1)

We have x approaching -1; in particular, this means x\neq-1, so that

\dfrac{2x^2-2}{x+1}=\dfrac{2(x-1)(x+1)}{x+1}=2(x-1)

Then

f'(-1)=\displaystyle\lim_{x\to-1}\frac{2x^2-2}{x+1}=\lim_{x\to-1}2(x-1)=2(-1-1)=-4

and the tangent line's equation is

y-f(-1)=f'(-1)(x-(-1))\implies y-4x-2

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