Question

I’m in AP Calculus AB. Can you help me find the limit?

Answer 1

0

Explanation:

Given the limit below:

`\lim _(h\rightarrow0)(\csc\left((\pi)/(2)+h\right)-\csc\left((\pi)/(2)\right))/(h)`

This limit is comparable to the derivative form below:

`\lim _(h\rightarrow0)(f\mleft(x+h\mright)-f(x))/(h)=f^(\prime)(x)`

Therefore:

`\lim _(h\rightarrow0)(\csc((\pi)/(2)+h)-\csc((\pi)/(2)))/(h)=(d)/(dx)\lbrack\csc ((\pi)/(2))\rbrack`

• The cosecant of a constant is a constant number.

,

• The derivative of a constant is 0.

Therefore:

`\begin{gathered} \lim _(h\rightarrow0)(\csc((\pi)/(2)+h)-\csc((\pi)/(2)))/(h)=(d)/(dx)\lbrack\csc ((\pi)/(2))\rbrack \\ =(d)/(dx)\lbrack1\rbrack \\ =0 \end{gathered}`

The limit is 0.