Find the exact value of the following limit: lim x approaches 0 e^6x... check photo above.

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asked Feb 25, 2022 207k views

1 Answer

Gabriel Esteban · Answer 1 · 2022-03-02T20:51:15+0000

Substituting x = 0 directly gives the indeterminate form 0/0, so you can use l'Hopital's rule (and you'll have to do that twice):

$\displaystyle\lim_(x\to0)(e^(6x)-6x-1)/(x^2)=\lim_(x\to0)(6e^(6x)-6)/(2x)=\lim_(x\to0)\frac{36e^(6x)}2$

The limand is continuous everywhere, so you can plug in x = 0 to get 36•1/2 = 18.