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Numerade Educator

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Problem 24 Easy Difficulty

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(3 + h)^{-1} - 3^{-1}}{h} $

Answer

$-\frac{1}{9}$

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Video Transcript

suppose you want to evaluate the limits of three plus age. Race two negative one minus three. Race the negative one over age as H approaches zero know that if we directly substitute H equals zero, we have three plus zero. Race two negative one minus three. Race the negative 1/0 which is just three. Race two negative one minus three. Race. The negative one over zero or that's just 0/0. Now this is indeterminate. That means we need to rewrite our function. And so from here we have the limit as H approaches zero of not that we can rewrite three plus age raised negative one and two one over three plus H minus the same way we have for three. Race two negative one that will be 1/3. This all over H. And then we combine the numerator. We get limit as H approaches zero of three minus three plus H over three times three plus age. This times the reciprocal of a trick is won over age. And then from here we simplify and we get the limit as a jew approaches zero of three minus three minus H over three times three plus age. This times one over age. And we get limit as H approaches zero of negative age over three times three plus age times one over age and know that we can cancel at the beach. And we have limit as H approaches zero of the negative one over three times three plus age and so substituting zero for age we have negative one over three times 340 This is just negative 1/9. And so this is the value of the limits.