Can a calculator find any limit?

Question

I am using CASIO fx-991ex and I noticed through my homework that when x approaches infinity or zero it's almost like using big or small numbers in calculator. For example $e^x$ while $x$ approaches $\infty$ is a very big number like using $e^9$ (+-) in the calculator, or $1/x$ while x approaches $0$ with $0.0001$ in the calculator (instead of $x$) it returns also a big number which I can assume it's $\infty$. So my question is when this thing doesnt work? I know that knowing how to find it out by myself is another important thing but im asking mainly for verifying purposes.

Answer 1

As a general principle it's fine to use a calculator as a tool to reinforce your intuition or guess an answer or confirm a theoretical calculation.

It's not a substitute for logical analysis.

There will be times when it "doesn't work" for several reasons. The calculator works with limited precision (just a finite number of decimal places) and finite range (there is a largest possible value). It may also fail when the limiting value is approached with oscillations rather than monotonically as in your two examples.

Answer 2

The Ackermann function is defined as:

$$A(0, n) = n+1$$ $$A(m, 0) = A(m-1, 1)$$ $$A(m, n) = A(m-1, A(m, n-1))$$

Define $A(n) = A(n, n)$.

The function $A$ grows extremely quickly. $A(3) = 61$. $A(4)$ is bigger than $10^{1355718576299609}$ - likely much bigger, but that is just the easiest lower bound estimate I got from Mathematica from the fact that computation overflows.

Define the inverse Ackermann function, $a(n)$, as the inverse of this function. So $a(61) = 3$. For any number your calculator can represent, the value of $a$ is less than $4$. Moreover, for any number Mathematica can represent on my machine, the value of $a$ is less than $4$. And yet the limit of $a(n)$ as $n \to \infty$ is $\infty$.

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Or consider $\sum_{n=1}^{\infty} \frac{1}{n}$. This is well-known to be infinite (by the Cauchy condensation test, for example). And yet $\sum_{n=1}^{1,000,000} \frac{1}{n}$ is approximately $14.3927$. If you use your calculator to estimate the value of the limit, you will get it wrong.

Answer 3

The process that you are using there tends to work all the time. It's exactly what I did in highschool when I knew nothing about limits. Like taking $x\to\infty$ I just typed $x=999999$ into my calculator and it gave me the correct limit. This was especially applicable when we were looking at the logistic equation for population growth. It worked similarly with taking $x\to 0$ as you said.