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Problem 32 Medium Difficulty

Evaluate the limit, if it exists.
$ \displaystyle \lim_{h \to 0}\frac{\frac{1}{(x + h)^2} - \frac{1}{x^2}}{h} $

Answer

$-\frac{2}{x^{3}}$

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Video Transcript

to evaluate this limit. We first do direct substitution. That means we replace each by zero and we get one over X plus zero squared minus one over X squared over zero. This gives us one over X squared minus one over X squared over zero or that's 0/0 An indeterminate value. This implies that we need to we write our function and so we have limit. As h approaches zero of we can combine the fractions in the numerator. That will be x squared minus X plus H squared over the common denominator, X plus H squared times X squared This times the reciprocal of age which is one over H. And then from here we can factor out the numerator and we get a limit As it approaches zero of we have x minus expose H times X plus expose each this all over X plus H squared times X squared Times one over age. And then from here we can simplify and we get the limit As it approaches zero of this one will be negative H. And then these times two x plus h over We have X plus H squared times X squared Times one over age. And then from here we can get rid of the age and we have limit As a church approaches zero of negative of two X plus H over we have X plus H squared times X squared. Now evaluating at zero. We get negative of two X plus zero over. We have explored zero squared times X squared which gives us negative two X over X squared times X squared dishes x rays to the fourth power. And you can simplify this further and we get negative two over X to the 3rd power. And so this is our limit.