Question

Evaluate the following limit, if it exists : limx→0 (12xe^x−12x) / (cos(5x)−1)

Answer 1

`\lim_(x \to 0) (12xe^x-12x)/(cos(5x)-1)=-(24)/(25)`

Explanation:

Notice that
`\lim_(x \to 0) (12xe^x-12x)/(cos(5x)-1)=(12(0)e^(0)-12(0))/(cos(5(0))-1)=(0)/(0)`, which is in indeterminate form, so we must use L'Hôpital's rule which states that
`\lim_(x \to c) (f(x))/(g(x))=\lim_(x \to c) (f'(x))/(g'(x))`. Basically, we keep differentiating the numerator and denominator until we can plug the limit in without having any discontinuities:

`(12xe^x-12x)/(cos(5x)-1)\\\\(12xe^x+12e^x-12)/(-5sin(5x))\\ \\(12xe^x+12e^x+12e^x)/(-25cos(5x))`

Now, plug in the limit and evaluate:

`(12(0)e^(0)+12e^(0)+12e^(0))/(-25cos(5(0)))\\ \\(12+12)/(-25cos(0))\\ \\(24)/(-25)\\ \\-(24)/(25)`

Thus,
`\lim_(x \to 0) (12xe^x-12x)/(cos(5x)-1)=-(24)/(25)`