Example (Non-Existent Limit): For $f(x) = \lceil x \rceil$ (ceiling function) at $x=1$:
LHL: $\lim_{x \to 1^-} \lceil x \rceil = 1$ (e.g., $\lceil 0.9 \rceil = 1$).
RHL: $\lim_{x \to 1^+} \lceil x \rceil = 2$ (e.g., $\lceil 1.1 \rceil = 2$).
Since $1 \neq 2$, $\lim_{x \to 1} \lceil x \rceil$ does not exist.

2.3. Algebra of Limits

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ (where $L, M$ are finite real numbers):

Sum/Difference: $\lim_{x \to a} [f(x) \pm g(x)] = L \pm M$
Product: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$ (if $M \neq 0$; if $M=0$, further analysis needed, e.g., $\pm \infty$ or DNE).
Constant Multiple: $\lim_{x \to a} [c \cdot f(x)] = c \cdot L$
Power: $\lim_{x \to a} [f(x)]^n = L^n$ (if $L^n$ is defined and real)

2.4. Techniques for Evaluating Limits

Direct Substitution: Applicable if $f(x)$ is continuous at $x=a$. Always the first attempt.
Eg: $\lim_{x \to 0} (e^x + \cos x) = e^0 + \cos 0 = 1 + 1 = 2$.
Factorization & Cancellation: For $\frac{0}{0}$ indeterminate forms. Factorize numerator/denominator to cancel the common factor $(x-a)$.
Eg: $\lim_{x \to -2} \frac{x^2 + 5x + 6}{x + 2} = \lim_{x \to -2} \frac{(x+2)(x+3)}{x+2} = \lim_{x \to -2} (x+3) = -2+3 = 1$.
Rationalization: For $\frac{0}{0}$ forms involving radicals. Multiply numerator and denominator by the conjugate.
Eg: $\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x} = \lim_{x \to 0} \frac{(\sqrt{x+4} - 2)(\sqrt{x+4} + 2)}{x(\sqrt{x+4} + 2)} = \lim_{x \to 0} \frac{x}{x(\sqrt{x+4} + 2)} = \lim_{x \to 0} \frac{1}{\sqrt{x+4} + 2} = \frac{1}{4}$.
Using Standard Limits: Apply specific formulas (see Page 2, Section 1 for a comprehensive list).
Eg: $\lim_{x \to 0} \frac{\sin(5x)}{3x} = \lim_{x \to 0} \frac{\sin(5x)}{5x} \cdot \frac{5x}{3x} = 1 \cdot \frac{5}{3} = \frac{5}{3}$.
Limits at Infinity: For rational functions $\frac{P(x)}{Q(x)}$, analyze by dividing all terms by the highest power of $x$ in the denominator, or by comparing degrees.
If deg($P$) < deg($Q$), limit is $0$.
If deg($P$) = deg($Q$), limit is ratio of leading coefficients.
If deg($P$) > deg($Q$), limit is $\pm \infty$.
Eg: $\lim_{x \to \infty} \frac{4x^3 + 2x^2 - 7}{9x^3 - x + 5} = \frac{4}{9}$.

3. Continuity: Unbroken Functions (Mathematics)

A function $f(x)$ is continuous at $x=a$ if its graph can be drawn without any breaks, holes, or jumps at that point. Three essential conditions must be satisfied:

$f(a)$ exists (the function is defined at $x=a$).
$\lim_{x \to a} f(x)$ exists (the limit from both sides converges to a single finite value).
$\lim_{x \to a} f(x) = f(a)$ (the limit value equals the function's actual value at $a$).

Types of Discontinuities:
- Removable (Hole): Occurs when $\lim_{x \to a} f(x)$ exists, but $f(a)$ is undefined or $f(a) \neq \lim_{x \to a} f(x)$.
- Jump: Occurs when LHL $\neq$ RHL. Common in piecewise functions.
- Infinite: Occurs when the function values approach $\pm \infty$ as $x \to a$ (vertical asymptote).
Physical Relevance: In physics and chemistry, most observable quantities (position, temperature, concentration) are typically modeled as continuous functions. Discontinuities often represent idealizations (e.g., instantaneous impacts) or specific physical phenomena (e.g., phase transitions in chemistry).

4. Derivatives: Instantaneous Rates of Change (M