MarkFL said:If $x=-1$, then you have:
\(\displaystyle \frac{-1}{1}=-1\)
The expression in the limit is one way to define the sign function:
\(\displaystyle \text{sgn}(x)\equiv\frac{x}{|x|}\) where \(\displaystyle x\ne0\)
This is equivalent to the piecewise definition:
\(\displaystyle \text{sgn}(x)\equiv\begin{cases}-1, & x<0 \\[3pt] 1, & 0<x \\ \end{cases}\)
tmt said:so $$\sqrt{ (-x)^2} = x$$?
The limit as x approaches negative infinity is a mathematical concept that represents the behavior of a function as the input (x) gets infinitely small (tends towards negative infinity). It describes the value that the function approaches or "approaches but never reaches" as x gets smaller and smaller.
The limit as x approaches negative infinity can be calculated by evaluating the function for extremely large negative values of x. Alternatively, one can use the concept of l'Hopital's rule, which states that the limit of a function can be found by taking the limit of the ratio of the derivatives of the function's numerator and denominator as x approaches negative infinity.
There are three possible outcomes when finding the limit as x approaches negative infinity: the limit may be a finite number, the limit may be positive or negative infinity, or the limit may not exist. The last outcome occurs when the function has different values from the left and right sides of the limit point.
The concept of limit as x approaches negative infinity is widely used in various fields, such as physics, engineering, and economics. It is used to model and predict the behavior of continuously changing systems, such as the speed of an object falling towards the ground or the rate of decay of a radioactive substance.
The main difference between the two concepts is the direction in which x is approaching infinity. In the limit as x approaches negative infinity, x gets smaller and smaller (tends towards negative infinity), while in the limit as x approaches positive infinity, x gets larger and larger (tends towards positive infinity). This difference can result in different limit values and behaviors of the function.