Use L'Hopital's rule to evaluate the following:(a) limx→1x3 + x2 − 2xx − 1(b) limx→∞x3e2x(c) limx→∞ln xx(d) limx→1+sin πx√x − 1Solution(a)limx→1x3 + x2 − 2xx − 1 , 00=⇒ limx→1x3 + x2 − 2xx − 1 = limx→13x2 + 2x − 21 = 3

Calculate the following limits:(a) limx→0sin(2x2 + x3)x ; (b) limx→0√x(sin(1/x) − cos(1/x)).

t's consider the limit given by:lim⁡𝑥→0−sin⁡(2𝑥)𝑥3⋅1−cos⁡(2𝑥)cos⁡(2𝑥)lim x→0 − ​ x 3 sin(2x)​ ⋅ cos(2x)1−cos(2x)​

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.lim x→1 3xx − 1 − 3ln(x)

The value of limx→∞ x3sin(1/x)−2x21+3x2 is __________

(a) limx→1x2 − 1x − 1