Question
lim x->pi cotx/cscx
Answers
GPT 3.5
To find the limit of (cotx/cscx) as x approaches pi, we can use the concept of L'Hopital's rule. L'Hopital's rule states that if we have a limit of the form f(x)/g(x), where both f(x) and g(x) approach 0 or infinity as x approaches the given value, then the limit of f(x)/g(x) as x approaches the given value is equal to the limit of f'(x)/g'(x) as x approaches the given value, provided this limit exists.
To apply L'Hopital's rule, we can rewrite the expression as (cosx/sinx)/(1/sinx). Now, as x approaches pi, sinx approaches 0 and cosx approaches -1. Thus, we have (-1/0)/(1/0), which is an indeterminate form of -∞/∞.
To find the limit, we can differentiate the numerator and denominator separately. The derivative of cosx is -sinx, and the derivative of sinx is cosx. So, the limit becomes (-(-sinx))/cosx = sinx/cosx = tanx.
Thus, the limit of (cotx/cscx) as x approaches pi is equal to tan(pi).
To apply L'Hopital's rule, we can rewrite the expression as (cosx/sinx)/(1/sinx). Now, as x approaches pi, sinx approaches 0 and cosx approaches -1. Thus, we have (-1/0)/(1/0), which is an indeterminate form of -∞/∞.
To find the limit, we can differentiate the numerator and denominator separately. The derivative of cosx is -sinx, and the derivative of sinx is cosx. So, the limit becomes (-(-sinx))/cosx = sinx/cosx = tanx.
Thus, the limit of (cotx/cscx) as x approaches pi is equal to tan(pi).