Given,

`lim_(x->-oo)` cschx

`= lim_(x->-oo) (2/(e^x -e^(-x)))`

as x -> to negative infinity ` e^x -> 0 ` and `e^-x` tends to negative infinity

so,

`lim_(x->-oo) (2/(e^x -e^(-x))) = lim_(x->-oo) (2/(0 -(-oo))) = 0`

Given,

`lim_(x->-oo)` cschx

`= lim_(x->-oo) (2/(e^x -e^(-x)))`

as x -> to negative infinity ` e^x -> 0 ` and `e^-x` tends to negative infinity

so,

`lim_(x->-oo) (2/(e^x -e^(-x))) = lim_(x->-oo) (2/(0 -(-oo))) = 0`