The Big Idea
One of the most important discoveries students make when studying calculus is that a limit value and a function value are not necessarily the same thing.
A function can approach one value while having a completely different value at the point itself.
This idea often feels strange at first because, in everyday arithmetic, a value is simply a value. Calculus introduces a new perspective: sometimes what happens near a point matters more than what happens at the point.
A Common Example
Consider the function:
For all values close to 1, the function behaves exactly like 
Therefore,
However, the function value at the point is:
Notice that:
- The limit is 1.
- The function value is 100.
Both statements are true simultaneously.
Why Does This Happen?
A limit examines the behavior of a function near a point.
When evaluating:
the values of 
The actual value 
In other words, limits focus on the journey, not necessarily the destination.
Applying Limit Laws
Consider the information:
and
Suppose it is also known that:
Now evaluate:
![\lim_{x\to1}[2f(x)+g(x)]^2](https://s0.wp.com/latex.php?latex=%5Clim_%7Bx%5Cto1%7D%5B2f%28x%29%2Bg%28x%29%5D%5E2&bg=ffffff&fg=000&s=0&c=20201002)
Using the limit laws:
![\lim_{x\to1}2f(x)+g(x)]^2=\left(2\lim_{x\to1}f(x)+\lim_{x\to1}g(x)\right)^2](https://s0.wp.com/latex.php?latex=%5Clim_%7Bx%5Cto1%7D2f%28x%29%2Bg%28x%29%5D%5E2%3D%5Cleft%282%5Clim_%7Bx%5Cto1%7Df%28x%29%2B%5Clim_%7Bx%5Cto1%7Dg%28x%29%5Cright%29%5E2&bg=ffffff&fg=000&s=0&c=20201002)
Substituting the known limits:
![\lim_{x\to1}[2f(x)+g(x)]^2=\left(2(3)+7\right)^2=13^2=169](https://s0.wp.com/latex.php?latex=%5Clim_%7Bx%5Cto1%7D%5B2f%28x%29%2Bg%28x%29%5D%5E2%3D%5Cleft%282%283%29%2B7%5Cright%29%5E2%3D13%5E2%3D169&bg=ffffff&fg=000&s=0&c=20201002)
Notice that the value:
plays no role in the calculation because limits depend on the behavior of the function near the point rather than the value at the point itself.
Can Exceptional Values Be Ignored?
This observation leads to an important idea in calculus.
Suppose a function is continuous almost everywhere except for a few isolated points where different values have been assigned.
In many calculus applications, the limit values provide a better description of the function’s overall behavior than those isolated exceptions.
For example:
The graph behaves like
A mathematician would often say that the function is “essentially” behaving like 
Removable Discontinuities
A discontinuity is called removable when the limit exists but the function value differs from that limit.
In the previous example:
but
The discontinuity can be removed by redefining the function value:
After doing so, the function becomes continuous.
Why Calculus Prefers Limits
Calculus focuses on:
- Continuity
- Rates of change
- Derivatives
- Areas under curves
- Long-term behavior
In many of these situations, changing a function at a single point has no effect on the overall result.
For example, the area under a curve remains unchanged whether:
or
A single point contributes no area.
This is why limits play such a central role in calculus.
A Helpful Mental Model
A useful way to think about functions is to imagine two layers:
Layer 1: Overall Behavior
How the graph behaves as values move through the domain.
Layer 2: Exceptional Point Values
Special values assigned at specific points.
Limits study Layer 1.
Function values study Layer 2.
When only a few exceptional points exist, calculus is often more interested in the overall behavior represented by the limits.
Key Takeaways
- A limit value and a function value are different concepts.
- The limit depends on values near a point.
- The function value depends on the exact point.
- A limit can exist even when the function value is different.
- Isolated exceptional values often do not affect the overall behavior of a function.
- Removable discontinuities occur when a limit exists but the function value does not match the limit.
Most importantly:
in general.
They are equal only when the function is continuous at the point.
Understanding this distinction is one of the foundational ideas that makes calculus possible.
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