To say that
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means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p.
The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.
Limits for general functions
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Limits of general functions
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Notable special limits
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Logarithmic and exponential functions
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Trigonometric functions
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