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17Calculus - Limit Key

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Limit Key

One key that you need to remember about limits is, when you use the limit notation \[ \lim_{x \rightarrow c}{ ~f(x) } \] this means that \(x\) APPROACHES \(c\) but IS NEVER EQUAL TO \(c\). That is, \(x\) can get as close as it wants to \(c\) but it will never actually equal \(c\). That seems simple enough but it is extremely important to remember.

However, there are many times when you can determine the limit by substituting x for c to calculate f(c). However, those are special cases that require special conditions and is not true in every case.

Here is an explanation of how this concept is written mathematically. Look more carefully at the definition of the limit at the top of the page. Notice that it requires \( \delta \gt 0 \). This means that \( |x-c| > 0 \) and, therefore, x can never equal c.

I know this seems like a minor point, but it isn't. If you remember this, you will have a good start on your way to understanding limits.

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