SOLUTION: Using the fact that lim x-->0 (sinx/x)=1 to find lim x-->0 ((1-cosx)/xsinx)) .

Algebra ->  Equations -> SOLUTION: Using the fact that lim x-->0 (sinx/x)=1 to find lim x-->0 ((1-cosx)/xsinx)) .      Log On


   



Question 1182300: Using the fact that lim x-->0 (sinx/x)=1 to find lim x-->0 ((1-cosx)/xsinx)) .
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Using the fact that lim x-->0 (sinx/x)=1 to find lim x-->0 ((1-cosx)/xsinx))
matrix%282%2C1%2Clim%2C%22x-%3E0%22+%29%28%281-cos%28x%29%29%2F%28x%2Asin%28x%29%29%29%22%22=%22%22

That given limit is 1 and we can multiply anything by 1 without changing its
value

matrix%282%2C1%2Clim%2C%22x-%3E0%22+%29%28%281-cos%28x%29%29%2F%28x%2Asin%28x%29%29%29%22%22%2A%22%22matrix%282%2C1%2Clim%2C%22x-%3E0%22+%29%28sin%28x%29%2Fx%29%22%22=%22%22

The product of limits equals the limit of products as long as they approach
the same number.

matrix%282%2C1%2Clim%2C%22x-%3E0%22+%29%28%28%281-cos%28x%29%29%2F%28x%2Asin%28x%29%29%29%28sin%28x%29%2Fx%29%5E%22%22%29%22%22=%22%22

matrix%282%2C1%2Clim%2C%22x-%3E0%22+%29%22%22=%22%22

matrix%282%2C1%2Clim%2C%22x-%3E0%22+%29%28%281%5E%22%22-cos%28x%29%29%2Fx%5E2%29%22%22=%22%22

%22%22=%22%22

%22%22=%22%22

%22%22=%22%22

%22%22=%22%22

%22%22=%22%22

%22%22=%22%22

The limit of a product equals the product of limits.

%22%22%2A%22%22%22%22=%22%22

In the left parentheses, the limit of a square is a square of the limit.
On the right parentheses, we can substitute what x approaches as long as it
gives a defined value, (not 0/0):

%22%22%2A%22%221%2F%281%2Bcos%280%29%29%22%22=%22%22

%281%5E%22%22%29%5E2%22%22%2A%22%221%2F%281%2B1%29%22%22=%22%22

1%22%22%2A%22%221%2F2%22%22=%22%22

1%2F2      <-- answer

Edwin