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Known Limits | Euler’s Limit Formula

The limits below are known limits, which can usually be used in limit calculations exercises without proving them.

\lim _ { x \rightarrow 0 } \frac{e^{x} - 1}{x} = 1

\lim _ { x \rightarrow 0 } \frac{\sin x}{x} = 1

Euler’s Limits Formulas

The following two limits are mathematically equivalent:

\lim _ { x \rightarrow 0 } ( 1 + x ) ^ {\frac{1}{x}} = e

\lim _ { x \rightarrow \infty } ( 1 + \frac {1}{x} ) ^{x} = e

Important explanation: Note that one must not reach exactly a limit as it appears on the list. It is sufficient to reach an expression that appears like the limit, but with another expression than x. However, in such a case, one must make sure that the expression tends to the same value x tends to. For examples of this explanation, please press here.

Also, notice that the limits shown here were proven in your course, in lecture or practice. I so, most likely you can use the results in problem-solving.

Press here for exercises and solutions using these limits

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