Limits and Continuity – Core Concepts and Applications

Definitions

Limit of a Function

For a real‑valued function $f$ defined on an interval containing $a$ (except possibly at $a$ itself), we say
$$\underset{x\to a}{\lim }f(x)=L$$
if for every $\epsilon >0$ there exists $\delta >0$ such that
$$0<|x-a|<\delta ⟹|f(x)-L|<\epsilon .$$
This is the epsilon‑delta definition of a limit.

One‑Sided Limits

The right‑hand limit $\underset{x\to a^{+}}{\lim }f(x)$ considers $x>a$ approaching $a$; the left‑hand limit $\underset{x\to a^{-}}{\lim }f(x)$ considers $x<a$.
If both one‑sided limits exist and are equal, the two‑sided limit exists.

Continuity

A function $f$ is continuous at $a$ if

1. $f(a)$ is defined,
2. $\underset{x\to a}{\lim }f(x)$ exists, and
3. $\underset{x\to a}{\lim }f(x)=f(a)$.

Types of Discontinuities

Limit Laws

Epsilon‑Delta Proof Example

Problem: Prove $\underset{x\to 2}{\lim }(3x+1)=7$.

Solution:
Let $\epsilon >0$. We need $|(3x+1)-7|<\epsilon$.
$$|(3x+1)-7|=|3x-6|=3|x-2|.$$ Choose $\delta =\frac{\epsilon }{3}$.
If $0<|x-2|<\delta$, then
$$|(3x+1)-7|=3|x-2|<3\delta =3\frac{\epsilon }{3}=\epsilon .$$ Thus the epsilon‑delta condition holds, proving the limit.

Continuity of Specific Functions

Example 1: $f(x)=\sqrt{9-x^2}$ on $[-3,3]$

Domain: $-3\le x\le 3$.
At interior points, $f$ is a composition of continuous functions (square, subtraction, square root).
At endpoints, ${\lim }_{x\to -3^{+}}f(x)=0$ and $f(-3)=0$; similarly for $x=3$.
Hence $f$ is continuous on $[-3,3]$.

Example 2: $f(x)=\frac{x^2-1}{x-1}$ at $x=1$

Simplify: $f(x)=x+1$ for $x\ne 1$.
Limit as $x\to 1$ is $2$, but $f(1)$ is undefined.
Thus $f$ has a removable discontinuity at $x=1$.

Worked Example – Local Extrema

Problem: Find the local maximum and minimum of
$$f(x)=4x^3-6x^2-9x+17.$$

Solution:

1. Compute derivative:
   $$f^{\prime }(x)=12x^2-12x-9.$$
2. Set $f^{\prime }(x)=0$:
   $$12x^2-12x-9=0⟹4x^2-4x-\frac{3}{4}=0.$$
   Solve using quadratic formula:
   $$x=\frac{4\pm \sqrt{16+12}}{8}=\frac{4\pm \sqrt{28}}{8}=\frac{4\pm 2\sqrt{7}}{8}=\frac{1\pm \sqrt{7}/2}{2}.$$
   Numerically:
   $x_1\approx -0.322$, $x_2\approx 1.822$.
3. Second derivative test:
   $$f^{\prime \prime }(x)=24x-12.$$
   Evaluate:
   $f^{\prime \prime }(x_1)=24(-0.322)-12\approx -20.33<0$ → local maximum.
   $f^{\prime \prime }(x_2)=24(1.822)-12\approx 32.33>0$ → local minimum.
4. Compute function values:
   $f(x_1)\approx 4(-0.322{)}^3-6(-0.322{)}^2-9(-0.322)+17\approx 19.2$.
   $f(x_2)\approx 4(1.822{)}^3-6(1.822{)}^2-9(1.822)+17\approx 5.4$.

Result:
Local maximum at $(x_1,f(x_1))\approx (-0.322,19.2)$.
Local minimum at $(x_2,f(x_2))\approx (1.822,5.4)$.

Real‑World Tie‑In

Consider a delivery company whose cost per kilometer is modeled by $C(d)=4d^3-6d^2-9d+17$ (in thousands of rupees).

• The local maximum cost occurs at $d\approx 0.32$ km, indicating a high fixed cost for very short trips.
• The local minimum at $d\approx 1.82$ km shows an optimal distance where marginal cost is lowest, guiding route planning.

Limit at Infinity

Problem:
$$\underset{h\to \infty }{\lim }\frac{\sqrt{6h+25}-5}{h}.$$

Solution:
Multiply numerator and denominator by the conjugate:
$$\frac{\sqrt{6h+25}-5}{h}\cdot \frac{\sqrt{6h+25}+5}{\sqrt{6h+25}+5}=\frac{6h+25-25}{h(\sqrt{6h+25}+5)}=\frac{6h}{h(\sqrt{6h+25}+5)}=\frac{6}{\sqrt{6h+25}+5}.$$
As $h\to \infty$, $\sqrt{6h+25}\sim \sqrt{6h}\sim \sqrt{6}\sqrt{h}$.
Thus the denominator grows without bound, so the limit is $0$.

Comparison Tables

One‑Sided vs Two‑Sided Limits

Continuity Conditions

Advantages and Disadvantages

In the real world

1. eSewa dynamic pricing – eSewa adjusts transaction fees based on real‑time demand curves, which are continuous functions of user load. The system uses limit concepts to predict fee changes as load approaches critical thresholds.
2. Daraz order queue – Daraz models the average waiting time of orders using queue theory. The long‑run average waiting time is a limit of the ratio of total waiting time to number of orders as the number of orders tends to infinity.
3. Kathmandu traffic flow – Traffic engineers model vehicle density as a continuous function of road position. The speed‑density relationship uses limits to estimate speed as density approaches maximum capacity, informing congestion mitigation strategies.

Exam tip

• Know the definitions: Be able to state the epsilon‑delta definition, one‑sided limits, and continuity criteria verbatim.
• Apply limit laws quickly: Practice simplifying expressions before applying laws; this saves time.
• Work through derivative‑based extrema problems: Remember to check the second derivative or use sign charts.
• Draw graphs: Even a quick sketch of the function’s behavior near the point of interest can reveal discontinuities or continuity.
• Use the “check‑list”: For continuity, verify (i) definition, (ii) existence of limit, (iii) equality.

Good luck!