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Two Important Limits Calculator

Numerically illustrate lim(x→0) sinx/x = 1 and lim(x→∞)(1+1/x)^x = e by entering x.

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lim(x→0) sinx/x = 1 lim(x→∞) (1+1/x)^x = e
First at x→0, second at x→∞
Small x for the first; large x for the second

📖 Tutorial｜Two Important Limits

1. Definition
Two important limits: ① lim(x→0) sinx/x = 1; ② lim(x→∞)(1+1/x)^x = e. They underlie many trigonometric and exponential limits.
2. Symbols
SymbolMeaning
sin x / xFirst important limit
(1+1/x)^xSecond important limit
eEuler’s number ≈ 2.71828
xVariable
3. How it works
  • The tool evaluates the expression at the given x;
  • For the first, take a tiny x (e.g. 0.001) to approach 1;
  • For the second, take a huge x (e.g. 100000) to approach e.
4. Steps
  1. Choose which important limit from the dropdown;
  2. Enter x: small for the first, large for the second;
  3. Click Calculate to watch the approach.
5. Example
Example: ① at x=0.001, sinx/x≈0.9999998→1; ② at x=100000, (1+1/x)^x≈2.71827→e.
6. Pitfalls
The first limit requires x→0 and the form sinx/x;
The second is a 1^∞ indeterminate form, not simply 1;
Very small x can add floating-point noise.

❓ FAQ｜Two Important Limits

Why are these limits important?
They are the building blocks for trigonometric and exponential-type limits; many problems reduce to them by substitution.
How is the second limit related to e?
e is defined as lim(x→∞)(1+1/x)^x ≈ 2.71828, the base of the natural logarithm.
Why isn’t my result exactly e for large x?
It is a finite approximation of (1+1/x)^x; the larger x the closer to e. At x=100000 it is accurate to about 5 digits.
Can I just set sinx/x = 1?
Only in the limiting sense x→0; never at a finite x.
Does the first limit generalize?
Yes. If u→0 then sinu/u→1; e.g. sin(2x)/(2x)→1 as x→0.