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` .
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