Lesson 1.2: Calculus (Limits, Continuity, Differentiation, Integration)

Calculus is one of the most important parts of Engineering Mathematics. It is widely asked in GATE, ESE, and PSU exams because it directly applies in fluid mechanics, thermodynamics, structural mechanics, heat transfer, and numerical methods. The main topics are Limits, Continuity, Differentiation, and Integration.

🔹 1. Limits

Definition:

lim⁡x→af(x)=L\lim_{x \to a} f(x) = Lx→alim​f(x)=L

means as $x\to a$, $f(x)\to L$.

Important Results:

• lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1limx→0​xsinx​=1

• lim⁡x→01−cos⁡xx2=12\lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}limx→0​x21−cosx​=21​

• lim⁡x→∞(1+1x)x=e\lim_{x \to \infty} (1 + \frac{1}{x})^x = elimx→∞​(1+x1​)x=e

L’Hospital’s Rule:

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)if 00 or ∞∞\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{if } \frac{0}{0} \text{ or } \frac{\infty}{\infty}x→alim​g(x)f(x)​=x→alim​g′(x)f′(x)​if 00​ or ∞∞​

🔹 2. Continuity

A function $f(x)$ is continuous at $x=a$ if:

lim⁡x→a−f(x)=lim⁡x→a+f(x)=f(a)\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)x→a−lim​f(x)=x→a+lim​f(x)=f(a)

Types of Discontinuities:

• Removable

• Jump

• Infinite

Example (GATE style):

f(x)={x2,x≥0−x,x<0f(x) = \begin{cases} x^2, & x \geq 0 \\ -x, & x < 0 \end{cases}f(x)={x2,−x,​x≥0x<0​

At $x=0$: both limits = 0 → Continuous.

🔹 3. Differentiation

Definition:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}f′(x)=h→0lim​hf(x+h)−f(x)​

Basic Rules:

• (xn)′=nxn−1(x^n)’ = n x^{n-1}(xn)′=nxn−1

• (ex)′=ex(e^x)’ = e^x(ex)′=ex

• (ln⁡x)′=1x(\ln x)’ = \frac{1}{x}(lnx)′=x1​

• Product Rule: (uv)′=u′v+uv′(uv)’ = u’v + uv’(uv)′=u′v+uv′

• Quotient Rule: (uv)′=u′v−uv′v2(\frac{u}{v})’ = \frac{u’v – uv’}{v^2}(vu​)′=v2u′v−uv′​

• Chain Rule: dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}dxdy​=dudy​⋅dxdu​

Applications in Engineering:

• Maxima & Minima → Optimization problems

• Slope of curve → Mechanical design

• Rate of change → Thermodynamics

🔹 4. Integration

Definition:

$$\int f(x)dx=F(x)+C$$

Standard Results:

• ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C

• ∫exdx=ex+C\int e^x dx = e^x + C∫exdx=ex+C

• $\int \sin xdx=-\cos x+C$

• $\int \cos xdx=\sin x+C$

Definite Integrals:

∫abf(x)dx=F(b)−F(a)\int_a^b f(x) dx = F(b) – F(a)∫ab​f(x)dx=F(b)−F(a)

Properties:

• ∫abf(x)dx=−∫baf(x)dx\int_a^b f(x) dx = – \int_b^a f(x) dx∫ab​f(x)dx=−∫ba​f(x)dx

• ∫aaf(x)dx=0\int_a^a f(x) dx = 0∫aa​f(x)dx=0

• If $f(x)$ is even: ∫−aaf(x)dx=2∫0af(x)dx\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx∫−aa​f(x)dx=2∫0a​f(x)dx

• If $f(x)$ is odd: ∫−aaf(x)dx=0\int_{-a}^a f(x) dx = 0∫−aa​f(x)dx=0

Applications in Engineering:

• Area under stress-strain curve

• Work done in thermodynamics

• Mass moment of inertia

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE 2020 ME):

lim⁡x→0sin⁡(3x)x=?\lim_{x \to 0} \frac{\sin(3x)}{x} = ?x→0lim​xsin(3x)​=?

👉 Solution: $=3$

Example 2 (PSU Exam):
If f(x)=x3−6×2+9x+2f(x) = x^3 – 6x^2 + 9x + 2f(x)=x3−6x2+9x+2, find maxima/minima.
👉 Solution:
f′(x)=3×2−12x+9=3(x−1)(x−3)f'(x) = 3x^2 – 12x + 9 = 3(x-1)(x-3)f′(x)=3x2−12x+9=3(x−1)(x−3)
Critical points: $x=1,3$

• At $x=1$, f′′(1)=−6<0f”(1) = -6 < 0f′′(1)=−6<0 → Max

• At $x=3$, f′′(3)=6>0f”(3) = 6 > 0f′′(3)=6>0 → Min

🔹 6. Practice Exercises

1. Evaluate: lim⁡x→01−cos⁡(2x)x2\lim_{x \to 0} \frac{1 – \cos(2x)}{x^2}limx→0​x21−cos(2x)​

2. Test continuity of f(x)=sin⁡xxf(x) = \frac{\sin x}{x}f(x)=xsinx​ at $x=0$.

3. Differentiate: y=e2xsin⁡xy = e^{2x}\sin xy=e2xsinx.

4. Find: ∫0πsin⁡2x dx\int_0^\pi \sin^2x \, dx∫0π​sin2xdx.

5. Solve: Find local maxima/minima of f(x)=x4−4x2f(x) = x^4 – 4x^2f(x)=x4−4x2.

🔹 7. Summary

• Limits: Value function approaches near a point.

• Continuity: No breaks in graph.

• Differentiation: Rate of change → slope, maxima, minima.

• Integration: Area under curve, work done, moments.

📌 Important Formula Sheet for GATE/PSU – Calculus

✅ Limits

• lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1limx→0​xsinx​=1

• lim⁡x→01−cos⁡xx2=12\lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}limx→0​x21−cosx​=21​

• lim⁡x→0tan⁡xx=1\lim_{x \to 0} \frac{\tan x}{x} = 1limx→0​xtanx​=1

• lim⁡x→∞(1+1x)x=e\lim_{x \to \infty} (1 + \frac{1}{x})^x = elimx→∞​(1+x1​)x=e

✅ Differentiation Rules

• (xn)′=nxn−1(x^n)’ = n x^{n-1}(xn)′=nxn−1

• (ln⁡x)′=1/x(\ln x)’ = 1/x(lnx)′=1/x

• (ex)′=ex(e^x)’ = e^x(ex)′=ex

• (ax)′=axln⁡a(a^x)’ = a^x \ln a(ax)′=axlna

• (sin⁡x)′=cos⁡x(\sin x)’ = \cos x(sinx)′=cosx

• (cos⁡x)′=−sin⁡x(\cos x)’ = -\sin x(cosx)′=−sinx

• (tan⁡x)′=sec⁡2x(\tan x)’ = \sec^2 x(tanx)′=sec2x

✅ Integration Rules

• ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C

• ∫exdx=ex+C\int e^x dx = e^x + C∫exdx=ex+C

• ∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln |x| + C∫x1​dx=ln∣x∣+C

• $\int \sin xdx=-\cos x+C$

• $\int \cos xdx=\sin x+C$

• ∫sec⁡2xdx=tan⁡x+C\int \sec^2 x dx = \tan x + C∫sec2xdx=tanx+C

• ∫csc⁡2xdx=−cot⁡x+C\int \csc^2 x dx = -\cot x + C∫csc2xdx=−cotx+C

• $\int \sec x\tan xdx=\sec x+C$

• $\int \csc x\cot xdx=-\csc x+C$

✅ Definite Integrals

• ∫0πsin⁡2xdx=π2\int_0^\pi \sin^2x dx = \frac{\pi}{2}∫0π​sin2xdx=2π​

• ∫0πcos⁡2xdx=π2\int_0^\pi \cos^2x dx = \frac{\pi}{2}∫0π​cos2xdx=2π​

• ∫−aaf(x)dx=0\int_{-a}^a f(x) dx = 0∫−aa​f(x)dx=0 if $f(x)$ odd

• ∫−aaf(x)dx=2∫0af(x)dx\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx∫−aa​f(x)dx=2∫0a​f(x)dx if $f(x)$ even