Mark M. answered • 07/04/17

Retired math prof. Calc 1, 2 and AP Calculus tutoring experience.

_{x→a}[(f(x) - f(a))/(x-a)]

^{2}, we have f'(3) = -1/9.

Min Q.

asked • 07/04/17lim

x-->3 (1/x-1/3)/(x-3)

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Mark M. answered • 07/04/17

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Retired math prof. Calc 1, 2 and AP Calculus tutoring experience.

Recall that f'(a) = lim_{x→a}[(f(x) - f(a))/(x-a)]

Letting a = 3 and f(x) = 1/x, the given limit is equal to f'(3).

Since f'(x) = -1/x^{2}, we have f'(3) = -1/9.

Arturo O. answered • 07/04/17

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You have what looks like an indeterminate form. In this case, it looks like 0/0. If possible, simplify the expression algebraically, and you might get a form that is not indeterminate. In this problem,

(1/x - 1/3) / (x - 3) = (3x / 3x)[(1/x - 1/3) / (x - 3)] = (3 - x) / [3x(x - 3)]

= -(x - 3) / [3x(x - 3)] = - 1 / 3x

This is not an indeterminate form, and you can substitute x = 3 and get -1/9. However, if you are left with and indeterminate form, you could use L'Hopital's rule, but that is not necessary in this problem

Michael J. answered • 07/04/17

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lim_{x-->3} (1/x-1/3)/(x-3)=lim_{x-->3} (3-x)/3x)/(x-3)=-lim_{x-->3}(1/3x)=-1/9

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Bobosharif S.

07/04/17