Maths limits?

If we are going to graph the function f(x) = x^2, we usually do it using small integers. Relative to those values of x, a number "close to zero" might be .01 or .001.

In some cases, we might be concerned with values of x that are in the neighborhood of 10^-6. Then, a value close to zero might be something on the order of 10^-9 or less. Many calculators start having trouble with numbers below about 10^-99, so a value of x around 10^-49 would be sufficiently "close to zero" to tax the calculator.

Some computers can represent numbers near 10^-4000 before they give underflow errors, so something near 10^-2000 would be effectively "close to zero."

In short, values "close to zero" are ones sufficiently small that you don't care about the difference from zero.

For purposes of discovering a function limit, "close to zero" is sufficiently small that the difference between the function value and its limit value is smaller than anything we care about. If you want to report the limit to 9 decimal places after the decimal point, then x=10^-5 is sufficient to make f(x) < 10^-9.

It means that the value of x is very close to 0.

Mathematically, it means that |x - 0| < δ, where δ is any arbitrary very small positive real number.