Question

Question #a5dc1

Answer 1

Use L'Hôpital's rule

Explanation:

Prove: `lim_(xto0)(sin(6x) + sin(8x))/(sin(4x) + sin(6x)) = 7/5`

The expression evaluated at the limit is the indeterminate form `0/0`, therefore, one should use L'Hôpital's rule

Take the derivative of the numerator:

`(d{sin(6x) + sin(8x)})/dx = 6cos(6x) + 8cos(8x)`

Take the derivative of the denominator:

`(d{sin(4x) + sin(6x)})/dx = 4cos(4x) + 6cos(6x)`

Make a new fraction with the new numerator and denominator:

`lim_(xto0)(6cos(6x) + 8cos(8x))/(4cos(4x) + 6cos(6x))`

The above can be evaluated at the limit:

`(6cos(6(0)) + 8cos(8(0)))/(4cos(4(0)) + 6cos(6(0))) = 14/10 = 7/5`

The rules says that this is the limit of the original expression:

`lim_(xto0)(sin(6x) + sin(8x))/(sin(4x) + sin(6x)) = 7/5`

Q.E.D.