Limits and Continuity

What a Limit Measures

A limit describes nearby behavior. In the notation $\lim_{x\to a}f(x)=L$, the input approaches $a$ while the output approaches $L$. The function value at the point is a separate issue: $f(a)$ may equal $L$, differ from $L$, or be undefined.

For example, consider

For inputs other than $1$, factor and simplify:

Therefore,

The original formula is undefined at $x=1$, but the limit still exists. This is a removable discontinuity.

When estimating from a table, use inputs increasingly close to the target from both sides, such as $a-0.1$, $a-0.01$, $a+0.01$, and $a+0.1$. A graph is interpreted in the same way: trace the output toward the target input from the left and from the right.

Takeaway: A limit concerns what happens near a point, not necessarily what happens at the point.

One-Sided and Two-Sided Behavior

A two-sided limit depends on agreement between the two directions of approach. The one-sided limit from the left is written $\lim_{x\to a^-}f(x)$, and the one-sided limit from the right is written $\lim_{x\to a^+}f(x)$. The relationship is

For the piecewise function

we obtain

Since the two values differ, $\lim_{x\to 3}f(x)$ does not exist. This is a jump discontinuity.

One-sided reasoning is especially important for piecewise definitions and for rational functions whose denominators approach zero. Always analyze each side before claiming that a two-sided limit exists.

Takeaway: A two-sided limit exists only when both one-sided limits agree.

Limit Laws and Algebraic Techniques

When the relevant limits exist, limits follow familiar algebraic laws. If $\lim_{x\to a}f(x)=L$ and $\lim_{x\to a}g(x)=M$, then

• $\lim_{x\to a}[f(x)+g(x)]=L+M$;

• $\lim_{x\to a}[f(x)-g(x)]=L-M$;

• $\lim_{x\to a}[f(x)g(x)]=LM$;

• $\lim_{x\to a}f(x)/g(x)=L/M$ when $M\ne0$;

• $\lim_{x\to a}[f(x)]^n=L^n$ for a positive integer $n$.

Direct substitution works for polynomials and for rational functions when the denominator does not approach zero. For example,

If direct substitution produces an indeterminate form such as $0/0$, continue with algebra. For

factor the numerator:

Thus,

Other methods include rationalizing with a conjugate, combining fractions, using known trigonometric limits, and applying the squeeze theorem.

Takeaway: Direct substitution is a first test, not a complete strategy when it produces an indeterminate form.

Infinite Limits and Vertical Asymptotes

An infinite limit describes unbounded function values near a finite input. For example,

The line $x=2$ is therefore a vertical asymptote. Infinity here describes unbounded growth; it is not an ordinary real number that the function reaches.

To determine signs near a denominator zero, inspect the factors. For

near $x=2$, the factors $x+1$ and $x+3$ are positive. The sign is controlled by $x-2$: it is negative from the left and positive from the right. Hence,

Do not confuse this situation with a limit at infinity. The expressions $\lim_{x\to a}f(x)=\pm\infty$ and $\lim_{x\to\pm\infty}f(x)=L$ describe different kinds of behavior.

Takeaway: Analyze factor signs on each side of a finite denominator zero to determine infinite behavior.

End Behavior and Horizontal Asymptotes

A limit at infinity describes end behavior as the input becomes arbitrarily large or arbitrarily negative. For a rational function

compare the degrees of the numerator and denominator:

• If the numerator degree is less than the denominator degree, the limit is $0$.

• If the degrees are equal, the limit is the ratio of leading coefficients, $a_n/b_m$.

• If the numerator degree is greater, there is no finite horizontal asymptote; polynomial division may reveal another asymptotic form.

For example,

Dividing by $x^3$ gives

As $x$ approaches infinity, the negative powers of $x$ approach zero, leaving $5/2$. Thus, $y=5/2$ is the corresponding horizontal asymptote.

Takeaway: Limits near a finite input describe local behavior; limits at infinity describe the ends of the graph.

Continuity at Points and on Intervals

To establish continuity at $x=a$, verify all three conditions:

1. $f(a)$ is defined.

2. $\lim_{x\to a}f(x)$ exists.

3. $\lim_{x\to a}f(x)=f(a)$.

Polynomials are continuous at every real input. Rational functions are continuous wherever their denominators are nonzero, and sums, products, quotients, and compositions preserve continuity wherever they are defined.

For a piecewise function, check the boundary separately. Let

The left-hand limit is

The right-hand limit and function value are both $2a+3$. Continuity requires

so $a=1$. With this value, both one-sided limits and the function value equal $5$.

On a closed interval $[a,b]$, continuity in the interior is combined with right-hand continuity at $a$ and left-hand continuity at $b$:

Takeaway: A convincing continuity justification checks definitions and boundary behavior rather than relying only on the appearance of a formula.

The Intermediate Value Theorem

The Intermediate Value Theorem applies when a function is continuous on an entire closed interval. If $f$ is continuous on $[a,b]$, then every number between $f(a)$ and $f(b)$ occurs at least once for some $c\in[a,b]$.

For root existence, it is enough to find opposite endpoint signs. Let

This polynomial is continuous on $[0,1]$, and

Because zero lies between the endpoint values, the theorem guarantees a number $c\in(0,1)$ such that

This proves that the equation $x^3+x-1=0$ has at least one solution in the interval. It does not prove that the solution is unique or provide its exact value.

The continuity hypothesis is essential. Also, endpoint values alone do not rule out an interior root: for $f(x)=(x-1)^2$ on $[0,2]$, both endpoint values are $1$, but $f(1)=0$.

A strong IVT argument should explicitly state the interval, verify continuity, calculate endpoint values, identify the intermediate value, and conclude only what the theorem guarantees.

Takeaway: The theorem guarantees existence under continuity, not uniqueness or an exact solution.

The Formal Epsilon-Delta Definition

The formal epsilon-delta definition makes the idea of approaching precise. The statement

means that for every $\varepsilon>0$, there exists a $\delta>0$ such that

The quantity $\varepsilon$ specifies the allowed output error, while $\delta$ specifies how close the input must be to $a$. The condition $0<|x-a|$ excludes the point itself because a limit concerns nearby inputs.

To prove

start with the output distance:

Choose

Then $0<|x-3|<\delta$ implies

Therefore the limit follows from the definition.

For a linear function $f(x)=mx+b$,

If $m\ne0$, choose $\delta=\varepsilon/|m|$; if $m=0$, any positive $\delta$ works. This confirms that every linear function is continuous everywhere.

Takeaway: In a formal proof, choose an input tolerance that forces the desired output tolerance.

A Reliable Problem-Solving Checklist

Use the following sequence to organize most limit and continuity problems:

1. Identify the approach: a finite input, a one-sided approach, or an infinite input.

2. Try direct substitution.

3. If the result is an indeterminate form such as $0/0$, simplify algebraically or use another appropriate limit method.

4. For one-sided behavior, analyze the two sides separately.

5. For rational functions at infinity, compare the numerator and denominator degrees or divide by the highest power of $x$.

6. For continuity, verify that the function is defined, that the limit exists, and that the limit equals the function value.

7. For the Intermediate Value Theorem, state continuity on the entire closed interval and evaluate the endpoints.

8. State precisely what has been proved, using “at least one” when uniqueness has not been established.

Common errors include treating $0/0$ as a limit value, assuming $f(a)$ determines the limit, checking only one side of a two-sided limit, confusing infinite limits with limits at infinity, and applying the Intermediate Value Theorem without verifying continuity.

Final checklist: Identify the type of question, select the relevant definition or theorem, verify its hypotheses, show the essential algebra, and phrase the conclusion with the correct level of certainty.