Engineering Mathematics Short Notes | Calulus – Functions of Single Variable | Limits, Continuity, Differentiability, MVT, Maxima–Minima, Integrals

ENGINEERING MATHEMATICS – FUNCTIONS OF SINGLE VARIABLE Short Notes (University And Gate Focus)

(Limit, Indeterminate Forms, L'Hospital, Continuity, Differentiability, MVT, Maxima–Minima, Taylor’s Theorem, Definite/Improper Integrals, Area & Volume)

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1. FUNCTIONS OF SINGLE VARIABLE – BASIC CONCEPTS

A function $f: D \to \mathbb{R}$ assigns each value $x \in D$ a unique real number $f(x)$.
Key points:

• Domain: set of allowable inputs

• Range: set of outputs

• Graph: visual representation $y = f(x)$

• Types: polynomial, rational, trigonometric, logarithmic, exponential

Understanding behavior (limits, continuity, differentiability) is foundational for calculus.

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2. LIMITS

A limit describes the value a function approaches as the input approaches some number.

2.1 Definition (ε–δ Not Required for Exam)

$$\lim_{x\to a} f(x) = L \quad \text{means} \quad f(x) \text{ approaches } L \text{ as } x \to a.$$

2.2 Basic Limit Laws

$$\lim (f+g) = \lim f + \lim g$$
$$\lim (f \cdot g) = (\lim f)(\lim g)$$
$$\lim \frac{f}{g} = \frac{\lim f}{\lim g} \quad (g \neq 0)$$

2.3 Standard Limits (Important for GATE)

$$\lim_{x\to 0} \frac{\sin x}{x} = 1$$
$$\lim_{x\to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$$ $$\lim_{x\to 0} (1+x)^{1/x} = e$$
$$\lim_{x\to 0} \frac{\tan x}{x} = 1$$
$$\lim_{x\to 0} \frac{e^x - 1}{x} = 1$$
$$\lim_{x\to 0} \frac{\ln(1 + x)}{x} = 1$$

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3. INDETERMINATE FORMS

Forms where limit cannot be evaluated directly:

• $\frac{0}{0}$

• $\frac{\infty}{\infty}$

• $0 \cdot \infty$
  

• $\infty - \infty$
  

• $0^0,{\mathrm{\infty }}^0,1^{\mathrm{\infty }}$

These require simplification or L'Hospital’s Rule.

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4. L'HOSPITAL'S RULE

Used when limit gives 0/0 or ∞/∞:

$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$

Apply repeatedly if form persists.

Examples

$$\lim_{x\to \infty} \frac{\ln x}{x} = 0$$

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5. CONTINUITY

A function is continuous at 

$x = a$ if:

1. $f(a)$ is defined

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

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

Types of Discontinuities

• Removable

• Jump

• Infinite

• Oscillatory

Continuity of Standard Functions

• Polynomial, exponential, logarithmic → always continuous

• Rational functions → continuous except where denominator = 0

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6. DIFFERENTIABILITY

A function is differentiable at ( x=a ) if derivative exists:

$$f^{\mathrm{\prime }}(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{\mathrm{h}}$$

Important Results

• Differentiability ⇒ Continuity

• Continuity ≠ Differentiability
  (Example: |x| at x = 0)

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7. MEAN VALUE THEOREMS

7.1 Rolle’s Theorem

If

1. f is continuous on [a, b]

2. f is differentiable on (a, b)

3. f(a) = f(b)

Then ∃ c ∈ (a, b):

                                                              f′(c)=0

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7.2 Lagrange’s Mean Value Theorem (LMVT)

If f is continuous on [a, b] and differentiable on (a, b):

$$f'(c) = \frac{f(b)-f(a)}{b-a}$$

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7.3 Cauchy Mean Value Theorem (CMVT)

For f and g satisfying LMVT conditions:

$$\frac{f^{\mathrm{\prime }}(c)}{g^{\mathrm{\prime }}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$

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8. MAXIMA & MINIMA

8.1 Critical Points

Solve:

$$f^{\mathrm{\prime }}(x)=0$$

8.2 Second Derivative Test

If $f'(a)=0$:

• $f^{\mathrm{\prime }\mathrm{\prime }}(a)>0$ local minimum

• $f^{\mathrm{\prime }\mathrm{\prime }}(a)<0$ local maximum

• $f^{\mathrm{\prime }\mathrm{\prime }}(a)=0$ further tests required

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9. TAYLOR’S THEOREM

For function f that is n-times differentiable:

$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$$

Special cases:

Taylor Series at x = 0 (Maclaurin Series)

1. $e^x = 1 + x + \frac{x^2}{2!} + \cdots$

2. $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
   

3. $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$

4. $\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$

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10. INTEGRALS

10.1 Fundamental Theorem of Calculus

If F is an antiderivative of f:

$$\frac{d}{dx} \int_a^x f(t)\,dt = f(x)$$
$$\int_a^b f(x)dx = F(b)-F(a)$$

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11. MEAN VALUE THEOREM FOR INTEGRALS

There exists 

$c \in [a,b]$:

$${\int }_a^{b}f(x)dx=f(c)(b-a)$$

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12. DEFINITE INTEGRALS

Properties

1. $\int_a^b f(x)dx = -\int_b^a f(x)dx$
   

2. $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$
   

3. If f is even:

   $$\int_{-a}^{a} f(x)dx = 2\int_0^a f(x)dx$$
   

4. If f is odd:

   $${\int }_{-a}^{a}f(x)dx=0$$

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13. IMPROPER INTEGRALS

Two types:

(i) Infinite Limits

$$\int_a^\infty f(x)dx = \lim_{b\to \infty}\int_a^b f(x)dx$$

(ii) Infinite Discontinuity

$$\int_a^b f(x)dx = \lim_{\epsilon\to 0}\int_{a}^{c-\epsilon} + \int_{c+\epsilon}^{b}$$

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14. APPLICATIONS OF DEFINITE INTEGRALS

14.1 Area Under Curve

$$A = \int_a^b f(x) dx$$

Area between curves:

$$A = \int_a^b |f(x) - g(x)| dx$$

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14.2 Area in Parametric Form

For $x = f(t), y = g(t)$

$$A = \int y \, dx = \int g(t)\,f'(t)\, dt$$

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14.3 Area in Polar Form

$$A=\frac{1}{2}{\int }_{{\theta }_1}^{{\theta }_2}{r}^{2}d\theta$$

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14.4 Volume of Solids of Revolution

(i) About x-axis (Disk Method)

$$V = \pi \int_a^b [f(x)]^2 dx$$

(ii) About y-axis

$$V = \pi \int_a^b [g(y)]^2 dy$$

(iii) Shell Method

$$V=2\pi {\int }_a^{b}xf(x)dx$$

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15. SOLVED EXAMPLES

Example 1: L'Hospital

$$\lim_{x\to 0} \frac{\ln(1+x)}{x} = 1$$

Example 2: Continuity

Check continuity of

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

After simplification:

$$f(x) = x+2, \ x\neq 2$$

Removable discontinuity at x = 2.

Example 3: Maximum

$$f(x)=x^3 -3x$$
$$f^{\mathrm{\prime }}(x)=3x^2-3=3(x^2-1)$$

Critical points: x = ±1.
Second derivative test gives max/min.

Example 4: Taylor Expansion

Expand $e^x$ about x=0 → standard series.

Example 5: Area

Find area between $y = x^2$ and x-axis from 0 to 2:

$$A={\int }_0^2{x}^{2}dx=\frac{8}{\mathrm{3}}$$

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16. SHORTCUTS FOR GATE

• Convert indeterminate form → apply L'Hospital.

• Even–odd property saves time in definite integrals.

• Improper integrals often converge if:

  $${\int }_1^{\mathrm{\infty }}\frac{1}{x^p}dx\text{ converges if }p>1$$

• Taylor expansion used for limits and approximations.

• Volumes → memorize disk & shell methods.

• MVT rarely asked directly, but conceptual traps appear.

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17. COMMON MISTAKES

• Applying L'Hospital to non-indeterminate forms.

• Confusing continuity with differentiability.

• Forgetting absolute value when computing area between curves.

• Ignoring convergence in improper integrals.

• Using MVT without checking continuity/differentiability.

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18. EXAM POINTERS

University Exams

• Write definitions + conditions clearly.

• Draw sketches for area/volume questions.

• Show full steps for Taylor expansion.

GATE

• Expect conceptual MCQs.

• Use series expansion for tricky limits.

• Volumes/areas often appear in applied contexts (physics-like).

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Keywords

• Functions of single variable notes

• Engineering mathematics notes

• GATE maths functions of single variable

• Limits and continuity notes

• Differentiability and mean value theorems

• Taylor’s theorem notes

• Maxima and minima short notes

• Definite and improper integrals

• Area and volume using integrals

• University exam maths notes