Question

Lim h ->0 (cos(pi+h) + 1 / h

Answer 1

Since
`\cos(\pi+0)+1=\cos(\pi)+1=-1+1=0`, the limit take on the indeterminate form 0/0. So, we can use the L'Hôpital's Rule:


`\lim_(h\to0)(\cos(\pi+h)+1)/(h)=\lim_(h\to0)(\sin(\pi+h)+0)/(1)\\\\ \lim_(h\to0)(\cos(\pi+h)+1)/(h)=\lim_(h\to0)\sin(\pi+h)\\\\ \lim_(h\to0)(\cos(\pi+h)+1)/(h)=\sin(\pi+0)\\\\ \lim_(h\to0)(\cos(\pi+h)+1)/(h)=\sin(\pi)\\\\ \boxed{\lim_(h\to0)(\cos(\pi+h)+1)/(h)=0}`