Dr. Math Azzam Platform | High School & University Mathematics

Review roadmap

Explanation

This review is designed as a full chapter rebuild, not a formula sheet. First identify what the question is testing, choose the method, justify the conclusion, and then check whether the answer is consistent with the graph or local behavior.

1.2 · See the limit

Tables, graphs, one-sided behavior, function value versus limit, and reasons a limit does not exist.

1.3 · Evaluate it

Direct substitution, limit laws, factoring, conjugates, special trigonometric limits, and the Squeeze Theorem.

1.4 · Decide continuity

Three continuity conditions, discontinuity types, piecewise and special functions, closed intervals, IVT, and bisection.

1.5 · Read unbounded behavior

Infinite limits, vertical asymptotes, factor-first classification, sign analysis, multiplicity, and trigonometric asymptotes.

Exam connection

Expect mixed questions: a graph can test limits and continuity at once; a rational expression can test factoring, a hole, and a vertical asymptote.

Mr. Azzam’s rule

Do not choose a technique because of how the expression looks. Substitute first, diagnose the result, then choose the next move.

Part A · Graphical, numerical, and one-sided limits

Explanation

Understand the idea before the notation

A limit asks where the nearby outputs are heading as x approaches a target. Read the left side and the right side, then compare them. If both approach the same height, that height is the two-sided limit—even if the actual point is missing or placed somewhere else.

Rule

Two-sided limit

\boxed{\lim_{x\to c}f(x)=L}\iff\lim_{x\to c^-}f(x)=L\;\text{and}\;\lim_{x\to c^+}f(x)=L

Rule

Limit versus function value

\lim_{x\to c}f(x)\text{ depends on nearby values, not necessarily }f(c)

Rule

Jump rule

\lim_{x\to c^-}f(x)\ne\lim_{x\to c^+}f(x)\Longrightarrow\lim_{x\to c}f(x)=\mathrm{DNE}

Rule

Approach from each side

x\to c^-:;x<c\qquad x\to c^+:;x>c

Interactive lab

Numerical limit challenge

Calculate the six outputs on either side of 4, then estimate the two-sided limit. The original function has no value at x = 4.

f(x)=\frac{x^2-16}{x-4}

x	Your f(x)	Feedback
3.9	____________	From the left
3.99	____________	From the left
3.999	____________	From the left
4.001	____________	From the right
4.01	____________	From the right
4.1	____________	From the right

Estimate the limit as x → 4

Factorization hint

x^2-16=(x-4)(x+4)

Cancel only when x ≠ 4. A limit describes nearby outputs, not necessarily a defined value at 4.

Common mistake. The value

f(c)

and the limit as

x\to c

are different questions. A function may be undefined at

c

while its limit exists.

Part B · Analytic limit toolbox

Key takeaway

Mr. Azzam’s decision routine

Substitute first. A real number means you are done. The indeterminate form 0/0 is not an answer; it tells you to simplify. Factor polynomial expressions, use a conjugate for radicals, and reshape trigonometric expressions to match a known special limit. Then substitute again.

Rule

Direct substitution

\boxed{\lim_{x\to c}f(x)=f(c)}\quad\text{when }f\text{ is continuous at }c

Rule

Indeterminate form

\frac{0}{0}\Longrightarrow\text{ simplify first, then substitute again}

Rule

Factoring identities

a^2-b^2=(a-b)(a+b)\qquad a^3-b^3=(a-b)(a^2+ab+b^2)

Rule

Conjugate identity

(a-b)(a+b)=a^2-b^2

Rule

Special trigonometric limits

\boxed{\lim_{x\to0}\frac{\sin x}{x}=1}\qquad\boxed{\lim_{x\to0}\frac{\tan x}{x}=1}

Rule

Squeeze Theorem

g(x)\le f(x)\le h(x),\;\lim g=\lim h=L\Longrightarrow\boxed{\lim f=L}

Method selection

\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}

Substitution gives $0/0$ . A radical is causing the problem, so use the conjugate rather than factoring blindly.

\frac{\sqrt{x}-3}{x-9}\cdot\frac{\sqrt{x}+3}{\sqrt{x}+3}=\frac1{\sqrt{x}+3}\to\frac16

Common mistake.

0/0

is not an answer. It is a diagnosis.

Part C · Continuity and discontinuities

Explanation

What continuity really means

At the point being tested, the function must have a value, the nearby behavior must approach one common value from both sides, and that approached value must equal the function value. If any one condition fails, the function is not continuous there.

Continuity at x = c

1.\ f(c)\text{ is defined (has a real value)}

2.\ \lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)

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

Classify the break

Removable: finite limit but a hole/wrong point. Jump: finite one-sided limits disagree. Infinite: at least one side is unbounded.

Piecewise breaking point

Ignore intervals where each formula is already continuous; test the point where the rule changes using LHL, RHL, and the assigned function value.

Closed interval [a,b]

Continuous inside

(a,b)

, right-continuous at

a

, and left-continuous at

b

.

Absolute value

Rewrite around the zero of the inside expression when a piecewise analysis is needed.

Greatest integer function

Expect jumps at integers. One-sided values around an integer are essential for deciding continuity.

Make a piecewise function continuous

f(x)=\begin{cases}2x+k,&x<3\\x^2-2,&x\ge3\end{cases}

For continuity at $3$ , match the two pieces: $6+k=7$ , so $k=1$ .

Part D · Intermediate Value Theorem and bisection

Explanation

Why IVT works

Picture a continuous curve drawn without lifting your pencil. If one endpoint is below a target y-value and the other is above it, the curve must cross that level somewhere between them. For a zero, the target is the x-axis.

Use IVT only after checking its hypotheses. If $f$ is continuous on $[a,b]$ and a number $N$ lies strictly between $f(a)$ and $f(b)$ , then some $c\in(a,b)$ satisfies $f(c)=N$ .

f(a)f(b)<0\quad\Rightarrow\quad\text{at least one zero in }(a,b)\quad\text{(when }f\text{ is continuous)}

Bisection · narrow the zero

f(x)=x^3+x-1,\qquad [0,1]

First verify the sign change. Then evaluate the midpoint. Keep the half-interval whose endpoint values still have opposite signs. Repeat to produce a smaller interval containing a zero.

Common mistake. IVT guarantees existence under its conditions; it does not automatically tell you the exact zero or guarantee that there is only one zero.

Part E · Infinite limits and vertical asymptotes

Explanation

Hole or vertical asymptote?

A hole has a finite nearby height after cancellation. A vertical asymptote has unbounded nearby behavior because a denominator factor remains. That is why we factor before classifying a denominator zero.

Meaning

+\infty

and

-\infty

describe unbounded behavior; they are not real-number limit values.

Vertical asymptote

x=c

is a VA when at least one one-sided limit at

c

is

+\infty

or

-\infty

.

Factor first

A denominator zero alone is not enough. Cancelled factor → investigate a hole. Factor remaining below → investigate a VA.

Sign routine

Near each candidate, determine the sign of every factor. The sign tells whether the branch goes to

+\infty

or

-\infty

.

Multiplicity shortcut

Even denominator power keeps the same sign across the zero; odd power changes sign. Always account for the numerator sign.

Trig asymptotes

Rewrite as quotients:

\tan x=\sin x/\cos x

and

\cot x=\cos x/\sin x

, then inspect denominator zeros.

One question · three skills

f(x)=\frac{(x-2)(x+4)}{(x-2)(x-1)^2}

At $x=2$ , the common factor cancels → removable discontinuity. At $x=1$ , an even denominator power remains → vertical asymptote with the same denominator sign on both sides; then use the numerator sign to determine the infinity direction.

Part F · Mixed chapter review

No lesson labels. Read the mathematics, identify the skill, then solve.

Practice

MIXED A · GRAPH

\text{Find }\lim_{x\to2}f(x),\qquad f(2),\qquad\text{and determine continuity at }x=2.

Practice

MIXED B · ANALYTIC LIMIT

\boxed{\displaystyle\lim_{x\to5}\frac{x^2-25}{x-5}}

1. Substitute first.

2. If you obtain

0/0

, choose the correct algebraic technique and evaluate the limit.

Practice

MIXED C · CONTINUITY PARAMETER

f(x)=\begin{cases}kx+2,&x<3\\x^2-1,&x\ge3\end{cases}

\text{Find }k\text{ so that }f\text{ is continuous at }x=3.

\lim_{x\to3^-}f(x)=\boxed{\phantom{000}}\qquad f(3)=\boxed{\phantom{000}}

Practice

MIXED D · IVT + BISECTION

f(x)=x^3+x-1,\qquad [0,1]

Step	a	b	midpoint	new interval
1
2

\text{First verify }f(0)f(1)<0,\text{ then complete two bisection steps.}

Practice

MIXED E · HOLE OR VERTICAL ASYMPTOTE?

f(x)=\frac{(x-2)(x+1)}{(x-2)(x+4)^2}

x=2

Hole or VA?

x=-4

Hole or VA?

\text{Factor}\;\longrightarrow\;\text{cancel common factors}\;\longrightarrow\;\text{classify each excluded value}

Practice

MIXED F · INFINITE LIMITS → SKETCH

\lim_{x\to-1^-}f(x)=+\infty,\qquad\lim_{x\to-1^+}f(x)=-\infty

\lim_{x\to3^-}f(x)=-\infty,\qquad\lim_{x\to3^+}f(x)=-\infty

Key takeaway

Exam habit

\boxed{\text{Read the mathematics}\;\to\;\text{identify the skill}\;\to\;\text{choose the rule}\;\to\;\text{solve}}

Mixed retrieval · choose and justify

Try these mixed checks without a lesson label. Identify the evidence and choose the method before opening a hint. Explanations are for review, not a recorded test score.

Your turn

Question 1 of 6

Suppose

\lim_{x\to2^-}f(x)=\lim_{x\to2^+}f(x)=5

but

f(2)=1

. Which conclusion is justified?

The limit is

1

The limit is

5

, but

f

is not continuous at

2

The limit does not existThe function is continuous because both sides agree

Review roadmap

1.2 · See the limit

1.3 · Evaluate it

1.4 · Decide continuity

1.5 · Read unbounded behavior

Exam connection

Mr. Azzam’s rule

Part A · Graphical, numerical, and one-sided limits

Understand the idea before the notation

Two-sided limit

Limit versus function value

Jump rule

Approach from each side

Part B · Analytic limit toolbox

Mr. Azzam’s decision routine

Direct substitution

Indeterminate form

Factoring identities

Conjugate identity

Special trigonometric limits

Squeeze Theorem

Part C · Continuity and discontinuities

What continuity really means

Continuity at x = c

Classify the break

Piecewise breaking point

Closed interval [a,b]

Absolute value

Greatest integer function

Part D · Intermediate Value Theorem and bisection

Why IVT works

Part E · Infinite limits and vertical asymptotes

Hole or vertical asymptote?

Meaning

Vertical asymptote

Factor first

Sign routine

Multiplicity shortcut

Trig asymptotes

Part F · Mixed chapter review

Mixed retrieval · choose and justify

Your turn

Your turn · student questions