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Answer:
since evaluating limit of the numerator and the denominator would result in an indeterminate form use L'Hopital's rule
then we get
calculate the derivatives
write all the numerators above the common denominator
simplify the complex fraction
use the L'Hopital's rule
calculate the derivatives
lim [-2sin(x)+2x /40x³]
x→0
reduce the fraction
lim [sin(x)-x/20x³]
x→0
use L'Hopital's rule
lim [d/dx(-sin(x)+x)/d/dx(20x³)]
x→0
lim [-cos(x)+1/60x²]
x→0
use the L'Hopital's rule
lim [d/dx(-cos(x)+1/d/dx(60x²)]
x→0
calculate the derivatives
lim (sin(x)/120x)
x→0
use L'Hopital's rule
lim [d/dx(sin(x))/d/dx(120x)]
x→0
now, we get
lim [cos(x)/120]
x→0
equate the limit
then , we get
=cos(0°)/120
=1/120
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