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maths1-week7

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Deep study for Quiz 2 week 7. Limits describe approaching behavior; continuity requires the approach to match the actual value.

Week map

Sequence terms → limit intuition → one-sided limits → two-sided limit → continuity at a point → types of discontinuity.

Limit notation

$\lim_{x \to a} f(x) = L$ → “limit of $f(x)$ as $x$ approaches $a$ is $L$ ” → value $f$ tends toward near $a$ , not necessarily at $a$ .
$\lim_{x \to a^-} f(x)$ → left-hand limit → approach from values less than $a$ .
$\lim_{x \to a^+} f(x)$ → right-hand limit → approach from values greater than $a$ .
$\lim_{x \to a} f(x)$ exists iff left and right limits exist and are equal.

Mini-example:

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

for

x \neq 1

. Near

x = 1

: factor to

x + 1

, so

\lim_{x \to 1} f(x) = 2

, even though

f(1)

is undefined.

Sequence limits

$\lim_{n \to \infty} a_n = L$ → terms $a_n$ get arbitrarily close to $L$ as $n$ grows.
For rational sequences, divide numerator and denominator by highest power of $n$ to guess limit.

Mini-example:

a_n = \frac{3n + 1}{n + 2}

. Divide top and bottom by

n

\frac{3 + 1/n}{1 + 2/n} \to 3

n \to \infty

Continuity notation

Continuous at $a$ : $\lim_{x \to a} f(x) = f(a)$ — limit exists, function defined, and they agree.
Removable discontinuity: limit exists but $f(a)$ missing or wrong — “hole” in graph.
Jump discontinuity: left and right limits exist but differ.
Infinite discontinuity: limit blows up to $\pm\infty$ .

Mini-example:

f(x) = |x|/x

x = 0

. Left limit

-1

, right limit

+1

— jump discontinuity.

Pattern families

Easy — Sequence pattern

Write first several terms from formula. Describe trend. Guess limit by table or simplifying for large

n

Medium — One-sided and two-sided limits

Evaluate limits from graphs. Identify left vs right behavior at breakpoints, absolute value corners, piecewise joins.

Hard — Continuity classification

Decide if function is continuous at a point. Identify discontinuity type. State what value would make function continuous (removable case).

Worked mini-examples

Example 1 — Direct substitution.

\lim_{x \to 3} (2x + 5) = 11

— polynomial, no issue.

Example 2 — Cancel factor.

\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4

Example 3 — One-sided.

f(x) = \begin{cases} x+1 & x < 0 \\ x^2 & x \geq 0 \end{cases}

. Left at 0:

1

; right at 0:

0

. Two-sided limit does not exist.

Example 4 — Sequence.

a_n = \frac{1}{n} \to 0

n \to \infty

Example 5 — Removable.

f(x) = \frac{\sin x}{x}

for

x \neq 0

f(0) = 0

\lim_{x \to 0} \frac{\sin x}{x} = 1 \neq f(0)

— removable if redefine

f(0) = 1

Traps

Assuming $\lim_{x \to a} f(x) = f(a)$ without checking.
Ignoring one-sided limits at piecewise boundaries.
Thinking limit at infinity means function equals that value.
Canceling $(x - a)$ without noting hole at $x = a$ .
Sequence limit confused with finite term value.

Diagnostic (try yourself)

Find $\lim_{x \to 5} (x^2 - 3x)$ .
Evaluate $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$ .
For $f(x) = \frac{|x|}{x}$ , what are the left and right limits at $x = 0$ ?
Sequence $a_n = \frac{2n}{n + 5}$ : what value does $a_n$ approach as $n \to \infty$ ?
A function has $\lim_{x \to 2} f(x) = 7$ but $f(2) = 3$ . Is it continuous at 2? What type of issue is this?

maths1-week7

Week map

Limit notation

Sequence limits

Continuity notation

Pattern families

Easy — Sequence pattern

Medium — One-sided and two-sided limits

Hard — Continuity classification

Worked mini-examples

Traps

Diagnostic (try yourself)

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