Why is limit of this function infinity as x approaches infinity
functionslimits
I typed this function into symbolab and I got the next answer:
Why is the limit of the function as x approaches infinity even possible to calculate if we get beyond the domain of the function? How does this online calculator process $\sqrt{-∞}$?
Best Answer
The online calculator is correct if you regard the complex plane as being completed by a single point $\infty$, but is wrong if you view $\infty$ as being a set of complex numbers with phase (as Mathematica does, for example).
The most unambiguous way to express the answer is probably "infinity, approached along the positive imaginary axis".
Dividing by really small numbers yields really big numbers. Consider for instance the reciprocal function $1/x$. As you plug in numbers for $x$ that are closer and closer to $0$, the outputs of $1/x$ will get bigger and bigger in size (possibly negative, if you're plugging in negative numbers). This occurs with all rational functions $f(x)/g(x)$: as $x$ approaches a value in which $g(x)$ is $0$ but $f(x)$ isn't, you get division by smaller and smaller numbers yielding bigger and bigger outputs.
The $y$ coordinate of a point on the graph is the output of the function applied to the $x$ coordinate, so if the outputs are getting bigger in size, graphically that means the points on the graph are going way up or way down.
Best Answer
The online calculator is correct if you regard the complex plane as being completed by a single point $\infty$, but is wrong if you view $\infty$ as being a set of complex numbers with phase (as Mathematica does, for example).
The most unambiguous way to express the answer is probably "infinity, approached along the positive imaginary axis".