How do you find the limit of $(x csc x + 1)/(x csc x)$ as x approaches 0?

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Answer 1 · 2016-01-29T06:08:49

$2$

Going into this, you should know the following limit identity:

$lim_(xrarr0)sinx/x=1$

This will come in handy.

The question rewritten is:

$lim_(xrarr0)(xcscx+1)/(xcscx)$

Split up the numerator.

$=lim_(xrarr0)1+1/(xcscx)$

$1/(xcscx)$ can be simplified since $1/cscx=sinx$ .

$=lim_(xrarr0)1+sinx/x$

Limits can be added to one another, as follows:

$=lim_(xrarr0)1+lim_(xrarr0)sinx/x$

Both of these limits are equal to $1$ , so the expression equals

$=1+1=2$

We can check a graph of the function. It should approach $2$ at $x=0$ .

graph{(xcscx+1)/(xcscx) [-14.04, 14.43, -1, 3]}

How do you find the limit of (x csc x + 1)/(x csc x) (x csc x + 1)/(x csc x) as x approaches 0?