Given,

`lim_(x->oo)` sech(x)

to find the value of `lim_(x->oo)`sechx

we need to find the value of

`lim_(x->-oo)` sechx and `lim_(x->+oo)`sechx

so,

the value of

`lim_(x->-oo)`sechx is as x tends to negative infinity the sech(x) -> 0

and similarly as

`lim_(x->+oo)`sechx is as x tends to positive infinity the sech(x) -> 0

So,

`lim_(x->-oo)`sechx`=lim_(x->+oo)` sechx`=0`

the limit exits for`lim_(x->oo)`sechx 

and the value is`lim_(x->oo)`sechx=0