`lim_(x->0^(-)) (cothx)` Find the limit

By definition, hyperbolic cotangent `coth(x)` is equal to `(cosh(x))/(sinh(x)),` which is equal to  `(e^x + e^(-x))/(e^x - e^(-x)).`

When `x -> 0^-,` both `e^x` and `e^(-x)` tend to `1.` But while `e^x -> 1^-,`  `e^(-x) = 1/e^x -> 1+.` Therefore the numerator tends to `1 + 1 = 2,` and the denominator tends to  `1^(-) - 1^+ = 0^-.`

And finally `2/0^-` gives the limit of `-oo.` This is the answer.