Question

find the limit algebraically (attached)

Answer 1

\[\lim_{x \rightarrow 0}\frac{ x \csc x+1 }{ x \csc x }\]

Answer 2

the way im doing it everything just cancels out -.- but im supposed to get 2 somehow

Answer 3

rewrite in terms of sine

Answer 4

\[\frac{x\csc(x)+1}{x\csc(x)}=\frac{\frac{x}{\sin(x)}+1}{\frac{x}{\sin(x)}}\]
\[=\frac{\frac{x+\sin(x)}{\sin(x)}}{\frac{x}{\sin(x)}}\]
\[=\frac{x+\sin(x)}{\sin(x)}\times \frac{\sin(x)}{x}\]
\[=\frac{x+\sin(x)}{x}=1+\frac{\sin(x)}{x}\]

Answer 5

simplify it first by doing this:

\[\lim_{x \rightarrow 0}\frac{ x \csc x }{ x \csc x } + \frac{1}{x cscx}\]

Answer 6

then take the limit, knowing tha t
\[\lim_{x\to 0}\frac{\sin(x)}{x}=1\] and you get \(1+1=2\)

Answer 7

actually @Hero method a lot shorter

Answer 8

Yeah, I'm all about shorter methods

Answer 9

and alternative methods too

Answer 10

thanks to both of you! haha i understand (: