Question

I don't know how do this, I know that I use L'hopitals Rule, I just don't know who to start.

Answer 1

Lim[x.sin(4π/x)] when x →∞. To apply the Hospital rule we need a fraction:

lim[x.sin(4π/x)] could be written:

lim [sin(4π/x)] / (1/x) . Now let's find the derivative of the numerator and the denominator:

Numerator = sin(4π/x) → (Numerator)' = cos(4π/x).(-4π/x²) [Chaine rule
(sinu)' = cosu. u'] So derivative of Numerator = cos(4π/x).(-4π/x²)

Denominator = 1/x → Numerator derivative = -1/x²

Now : (numerator)'/(denominator)' = cos(4π/x).(-4π/x²) / -1/x²
Simplify by x² : → cos(4π/x).(-4π) / -1

OR  cos(4π/x).(4π) . When x→∞ , 4π/x → 0 and cos(0) = 1, then:

lim[x.sin(4π/x)] when x →∞. is 4π