Question

Evaluate the limit using L'Hopital's rule

lim 4x⁵ cos ((13/x⁵) + (π/2))
x→[infinity]

Answer 1

To evaluate the given limit using L'Hopital's rule, we first find the derivatives of the numerator and denominator separately and then take the limit again. After multiple applications of L'Hopital's rule, the final result is +∞.

Step-by-step explanation:

L'Hopital's Rule to Evaluate the Limit:

To evaluate the limit using L'Hopital's rule, we need to find the derivative of both the numerator and denominator separately and take the limit again. Let's start by finding the derivatives:

Derivative of 4x⁵:
d(4x⁵)/dx = 20x⁴

Derivative of cos((13/x⁵) + (π/2)):
d(cos((13/x⁵) + (π/2)))/dx = -13(5/x⁵)²(-1/x⁶)

Now, let's simplify the expression and find the limit:

lim x→∞ (20x⁴) / (-13(5/x⁵)²(-1/x⁶))

Applying L'Hopital's rule again, we differentiate the numerator and denominator once more. Continuing this process until we can evaluate the limit:

lim x→∞ 80x³ / (13(5/x⁵)²(6/x⁷))

Simplifying further:

lim x→∞ 80x³ / (90/x⁸)

Finally, taking the limit:

lim x→∞ 80x¹¹ / 90 = +∞