Question

Lim x-->0 (sin5x)/(x/2)

Answer 1

lim x-->0 (sin5x)/(x/2) неопределенность 0/0

при x-->0 sin5x ~ 5x

lim x-->0 (5x)/(x/2) = lim x-->0 (5*2) = lim x-->0 10 = 10

Answer 2

To evaluate the limit:

lim (x --> 0) (sin 5x) / (x / 2)

We can simplify the expression by using the fact that sin(x) / x approaches 1 as x approaches 0. This is a well-known limit in calculus.

lim (x --> 0) (sin 5x) / (x / 2)

= lim (x --> 0) [(sin 5x) / (x)] * (2 / 1) // Dividing by x is the same as multiplying by 1/x

= 5 * lim (x --> 0) (sin 5x) / (5x) * (2 / 1) // Using the property of the limit of a constant times a function

Now, we can focus on the term:

lim (x --> 0) (sin 5x) / (5x)

We know that sin(x) / x approaches 1 as x approaches 0. So, we can replace sin(5x) / (5x) with 1:

lim (x --> 0) (sin 5x) / (5x) = 1

Now, putting it all together:

lim (x --> 0) (sin 5x) / (x / 2)

= 5 * lim (x --> 0) (sin 5x) / (5x) * (2 / 1)

= 5 * 1 * 2

= 10

Therefore, the limit of (sin 5x) / (x / 2) as x approaches 0 is 10.