Math101 · Independent Ontario learning library
Math101learn.math101.caLimits
A limit describes the value a function approaches as its input moves near a target, whether or not the function is defined there.
Limits let calculus discuss motion toward a point without requiring the function's value at that point to tell the whole story.
Meaning and notation
The statement
means that $f(x)$ can be made as close to $L$ as desired by taking $x$ sufficiently close to $a$ from both sides, without requiring $x=a$.
The limit concerns nearby behaviour. The actual value $f(a)$ may equal $L$, differ from $L$, or be undefined.
Estimating from a table
Choose inputs approaching $a$ from below and above. If the corresponding outputs approach the same value, that value is evidence for the limit.
Use enough precision to see a trend, but remember that a finite table suggests rather than proves a limit. A graph or algebraic argument should support the conclusion.
Estimating from a graph
Trace the graph toward $x=a$ from the left and right. Focus on the approached height, not merely a filled or open point at $x=a$.
An open circle can represent the limiting value even when the function is missing there. A filled point elsewhere at the same input gives $f(a)$ but does not overwrite the nearby approach.
Direct substitution and continuity
Polynomials and many familiar functions are continuous on their domains. If $f$ is continuous at $a$, then
Try direct substitution first. If it gives an ordinary real number, the limit is often finished. If it gives an indeterminate form such as $0/0$, simplify the expression before evaluating.
Worked example: remove a common factor
The original function has a hole at $x=3$, yet the limit exists.
One-sided limits
The left-hand limit is
and the right-hand limit is
A two-sided limit exists only when both one-sided limits exist and are equal. Different approach values at a jump mean the two-sided limit does not exist.
Infinite limits and asymptotes
If $f(x)$ grows without bound as $x$ approaches $a$, notation such as
describes the behaviour. Infinity is not a real limit value; it signals unbounded growth. A vertical asymptote often occurs at $x=a$.
The two sides may approach different infinities and should be checked separately.
Limits at infinity
The expression
describes end behaviour as inputs grow. For rational functions, compare leading degrees or divide by the highest relevant power. A finite value $L$ often corresponds to horizontal asymptote $y=L$.
The graph may cross a horizontal asymptote at finite inputs.
Limit laws
When component limits exist, limits pass through sums, differences, constant multiples, products, and quotients whose denominator limit is nonzero. Powers and roots also behave predictably on valid domains.
These laws justify substitution after an expression has been rewritten into continuous pieces.
Common mistakes
Substituting the point and stopping at $0/0$. It is an indeterminate signal to simplify.
Confusing $f(a)$ with the limit. Nearby behaviour may differ from the point value.
Checking only one side. A two-sided limit requires agreement from both sides.
Treating infinity as an ordinary number. It describes unbounded behaviour.
Cancelling terms instead of factors. Factor algebraically before cancellation.
Quick self-check
- What happens under direct substitution?
- If indeterminate, can I factor, rationalize, or combine fractions?
- Do left- and right-hand behaviours agree?
- Am I reporting an approached value rather than blindly reading $f(a)$?
- Does a graph or table support the algebra?
- Are domain restrictions and asymptotes handled explicitly?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Evaluate lim as x approaches 3 of (x² − 9)/(x − 3).
- (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3)
- For x ≠ 3, this is x + 3.
- The limit is 3 + 3 = 6.
End of lesson
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