Calculus Guide
Learn what limits mean, how to evaluate them graphically, numerically, and algebraically, and how limits form the foundation of calculus.
The Foundation of Calculus
A limit describes the value that a function approaches as its input approaches a particular number.
Limits allow us to study what happens near a point without requiring the function to actually equal that value at the point itself.
Limit Notation
This means that as \(x\) gets closer and closer to \(a\), the values of \(f(x)\) get closer and closer to \(L\).
Big Idea
When evaluating \( \displaystyle \lim\limits_{x \to a} f(x) \), we care about what happens to the function as \(x\) approaches \(a\) from nearby values.
The behavior of the function at \(x=a\) can be completely different from the limit.
An Important Distinction
One of the most important ideas in limits is understanding that the value of a function at a point and the limit at that point are not necessarily the same thing.
Function Value
This asks:
What is the actual value of the function when \(x=a\)?
Limit
This asks:
What value does the function approach as \(x\) approaches \(a\)?
Example
Suppose a graph approaches the point \((2,5)\) from both sides, but there is an open circle at \((2,5)\) and a filled point at \((2,1)\).
The function value and the limit are different, and that is perfectly possible.
Three Perspectives
Limits can be investigated in three main ways. Each method describes the same mathematical idea from a different perspective.
Graphically
Look at the \(y\)-value that the graph approaches as \(x\) moves toward the target \(x\)-value.
Pay attention to the graph from both the left and the right.
Numerically
Choose \(x\)-values closer and closer to the target value from both sides.
Then examine the corresponding values of \(f(x)\).
Algebraically
Use direct substitution, factoring, rationalizing, and limit laws to evaluate the limit exactly.
Numerical Example
| \(x\) | \(f(x)\) |
|---|---|
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 1.999 | 3.999 |
| 2.001 | 4.001 |
| 2.01 | 4.01 |
| 2.1 | 4.1 |
These values suggest that \[ \lim_{x\to2}f(x)=4. \]
Always Try This First
For many limits, the easiest first step is simply to substitute the value that \(x\) is approaching into the function.
First Move
Consider:
\[ \lim_{x\to3}(x^2+2x-1) \]Substitute \(x=3\):
\[ 3^2+2(3)-1 \] \[ =9+6-1 \] \[ =14 \]Important
If direct substitution gives you an ordinary real number, you are usually finished.
When Substitution Fails
Sometimes direct substitution produces \(\frac{0}{0}\).
This is called an indeterminate form.
Common Misunderstanding
Getting \(\frac{0}{0}\) does not mean the limit is zero.
It also does not automatically mean that the limit does not exist.
Instead, it tells us that the original expression must be simplified before the limit can be determined.
Algebraic Strategy
When substitution produces \(0/0\) and the expression contains polynomials, factoring is often the next step.
Example
Direct substitution gives:
\[ \frac{3^2-9}{3-3} = \frac{0}{0} \]Factor the numerator:
\[ x^2-9=(x-3)(x+3) \]Rewrite the limit:
\[ \lim_{x\to3} \frac{(x-3)(x+3)}{x-3} \]For \(x\neq3\), cancel the common factor:
\[ \lim_{x\to3}(x+3) \]Now substitute:
\[ 3+3=6 \]Why Can We Cancel?
The original function is not defined at \(x=3\), because its denominator becomes zero.
But a limit asks what happens near \(x=3\). For nearby values where \(x\neq3\), the expression is equivalent to \(x+3\).
Algebraic Strategy
When square roots produce a \(0/0\) indeterminate form, multiply by the conjugate.
Conjugates
Multiplying conjugates uses the difference-of-squares pattern:
\[ (a+b)(a-b)=a^2-b^2 \]Example
Direct substitution gives:
\[ \frac{0}{0} \]Multiply by the conjugate:
\[ \frac{\sqrt{x}-2}{x-4} \cdot \frac{\sqrt{x}+2}{\sqrt{x}+2} \]Multiply the numerator:
\[ (\sqrt{x}-2)(\sqrt{x}+2) = x-4 \]So:
\[ \lim_{x\to4} \frac{x-4} {(x-4)(\sqrt{x}+2)} \]Cancel \(x-4\):
\[ \lim_{x\to4} \frac{1}{\sqrt{x}+2} \]Substitute \(x=4\):
\[ \frac{1}{2+2} = \frac14 \]Combining Limits
Limit laws allow us to break complicated expressions into simpler pieces.
Suppose:
\[ \lim_{x\to a}f(x)=L \]and
\[ \lim_{x\to a}g(x)=M. \]Sum Law
\[ \lim_{x\to a} [f(x)+g(x)] = L+M \]Difference Law
\[ \lim_{x\to a} [f(x)-g(x)] = L-M \]Constant Multiple
\[ \lim_{x\to a} [cf(x)] = cL \]Product Law
\[ \lim_{x\to a} [f(x)g(x)] = LM \]Quotient Law
\[ \lim_{x\to a} \frac{f(x)}{g(x)} = \frac{L}{M} \]provided \(M\neq0\)
Power Law
\[ \lim_{x\to a} [f(x)]^n = L^n \]Root Law
\[ \lim_{x\to a} \sqrt[n]{f(x)} = \sqrt[n]{L} \]whenever the root is defined
Constant Law
\[ \lim_{x\to a}c=c \]Direction Matters
Sometimes we need to know how a function behaves when approaching a point from only one direction.
Left-Hand Limit
\(x\) approaches \(a\) using values less than \(a\).
Right-Hand Limit
\(x\) approaches \(a\) using values greater than \(a\).
Reading the Notation
The superscript does not indicate whether the answer is positive or negative.
means approach \(a\) from the left.
means approach \(a\) from the right.
Both Sides Must Agree
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
Requirement
\[ \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) = L \]Then:
\[ \boxed{ \lim_{x\to a}f(x)=L } \]Does Not Exist
Suppose:
\[ \lim_{x\to2^-}f(x)=3 \]while:
\[ \lim_{x\to2^+}f(x)=7. \]Because the two sides approach different values:
\[ \boxed{ \lim_{x\to2}f(x) \text{ DNE} } \]The Two-Sided Test
If they match, that common value is the two-sided limit. If they do not match, the two-sided limit does not exist.
Unbounded Behavior
Sometimes function values do not approach a finite number. Instead, they grow without bound.
Positive Infinity
\[ \lim_{x\to a}f(x)=\infty \]The function increases without bound as \(x\) approaches \(a\).
Negative Infinity
\[ \lim_{x\to a}f(x)=-\infty \]The function decreases without bound as \(x\) approaches \(a\).
Example
Consider:
\[ f(x)=\frac1x \]As \(x\) approaches zero from the right:
\[ \lim_{x\to0^+}\frac1x = \infty \]As \(x\) approaches zero from the left:
\[ \lim_{x\to0^-}\frac1x = -\infty \]The one-sided behaviors are different, so:
\[ \boxed{ \lim_{x\to0}\frac1x \text{ DNE} } \]Vertical Asymptote
If a function approaches \(+\infty\) or \(-\infty\) as \(x\) approaches \(a\) from at least one side, then
\[ x=a \]is typically a vertical asymptote of the graph.
Technical Note
Infinity is not a real number. Writing a limit as \(+\infty\) or \(-\infty\) describes unbounded behavior rather than a finite limit value.
The Next Connection
Limits give us a precise way to define whether a function is continuous at a point.
Continuous at \(x=a\)
A function \(f\) is continuous at \(x=a\) when all three of the following conditions are true.
Intuitive Picture
Informally, a continuous graph can be drawn through \(x=a\) without lifting your pencil, provided there is no hole, jump, or vertical asymptote there.
Watch Out
The function value and the limit may be different. Always distinguish between what happens at a point and what happens near it.
\(0/0\) is an indeterminate form. It signals that more algebra is needed.
You cannot cancel across addition or subtraction.
For a two-sided limit, the left-hand and right-hand limits must agree.
These symbols indicate the direction of approach, not whether the function value is negative or positive.
Infinity describes unbounded behavior. It is not an ordinary real-number output.
Always try direct substitution first. Many limits require no additional work.
Problem-Solving Roadmap
When you see an algebraic limit, use this sequence before trying anything more complicated.
Start Here
Plug the approaching value into the function.
If You Get a Number
That number is normally the limit.
If You Get \(0/0\)
Look for factoring, cancellation, or a conjugate.
Substitute Again
Once the troublesome factor is removed, try direct substitution again.
If Behavior Changes by Side
Compare the left-hand and right-hand limits.
Keep This Handy
Basic Limit
\[ \lim_{x\to a}f(x)=L \]\(f(x)\) approaches \(L\) as \(x\) approaches \(a\).
Left-Hand
\[ \lim_{x\to a^-}f(x) \]Approach from values less than \(a\).
Right-Hand
\[ \lim_{x\to a^+}f(x) \]Approach from values greater than \(a\).
Two-Sided Limit
\[ \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) \]Both sides must agree.
Indeterminate Form
\[ \frac00 \]Simplify before deciding the limit.
Continuity
\[ \lim_{x\to a}f(x)=f(a) \]Required for continuity at \(x=a\).
Infinite Limit
\[ \lim_{x\to a}f(x)=\infty \]Function values grow without bound.
Vertical Asymptote
\[ x=a \]Often occurs when the function becomes unbounded near \(x=a\).
Before You Practice
Your Turn
Practice evaluating limits with direct substitution, factoring, conjugates, one-sided limits, graphs, and continuity.
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