Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 18

A limit is the value that the function approach as x approaches "a".
In the given problem, the x-gt- oo indicates that independent variable x approaches large negative numbers for given function: f(x)=tanh(x) .
The function f(x)= tanh(x) is the hyperbolic tangent function. Its domain is all real number that can be expressed with the interval notation (-oo,oo) . It is a symmetric odd function. It also has an inflection point that can be found at x=0. There are no local extrema that can found in the continuous function of hyperbolic tangent.
The "attached image" is the graph of f(x)=tanh(x) .
By graphical inspection, as graph continues to left of y-axis or x approaches -oo , it approaches y = -1 .
Therefore, the limit will be:
lim_(x-gt-oo) [tanh(x)] = -1 .