question evaluate the following limit using lhospitals rule. lim x→0 -4x + 4 sin(x) / 11x³ enter an exact…

Explanation:

Step1: Check L'Hopital - rule condition

When (x\rightarrow0), (\frac{-4x + 4\sin(x)}{11x^{3}}) is in (\frac{0}{0}) form since (-4\times0+4\sin(0)=0) and (11\times0^{3}=0).

Step2: Differentiate numerator and denominator

The derivative of the numerator (y_1=-4x + 4\sin(x)) is (y_1^\prime=-4 + 4\cos(x)), and the derivative of the denominator (y_2 = 11x^{3}) is (y_2^\prime=33x^{2}). So the limit becomes (\lim_{x\rightarrow0}\frac{-4 + 4\cos(x)}{33x^{2}}), which is still in (\frac{0}{0}) form ((-4 + 4\cos(0)=-4 + 4=0)).

Step3: Differentiate again

Differentiate (y_1^\prime=-4 + 4\cos(x)) to get (y_1^{\prime\prime}=-4\sin(x)), and differentiate (y_2^\prime=33x^{2}) to get (y_2^{\prime\prime}=66x). The limit is now (\lim_{x\rightarrow0}\frac{-4\sin(x)}{66x}), which is in (\frac{0}{0}) form ((-4\sin(0)=0)).

Step4: Differentiate one - more time

Differentiate (y_1^{\prime\prime}=-4\sin(x)) to get (y_1^{\prime\prime\prime}=-4\cos(x)), and differentiate (y_2^{\prime\prime}=66x) to get (y_2^{\prime\prime\prime}=66).

Step5: Evaluate the limit

(\lim_{x\rightarrow0}\frac{-4\cos(x)}{66}=\frac{-4\cos(0)}{66}=\frac{-4\times1}{66}=-\frac{2}{33})

Answer:

(-\frac{2}{33})