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Limits and Continuity Notes for GATE CS

Limits and Continuity notes for GATE CS: 22 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

limits and continuity notes

Chapter Roadmap: Limits and Continuity

Chapter Journey: Limits & Continuity 01 Evaluation of Indeterminate Limits L'Hôpital's Rule, Standard Limits, Series Expansion. The core engine. 02 Continuity of Piecewise Functions Matching left-hand and right-hand limits. Finding unknown constants. 03 Discontinuities of Digit-Defined Functions Advanced logic-based continuity checks. Rare but high-difficulty. Goal: Master the algebraic manipulation of limits first. Everything else depends on it.

The Hero Concept: Why Indeterminate Forms?

What is an Indeterminate Form? A limit lim_{x \to a} \frac{f ( x )}{g ( x )} is indeterminate if direct substitution yields: \frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, \infty - \infty, 1^{\infty}, 0^{0}, \infty^{0} Numerator vs Denominator approaching zero If numerator approaches zero faster, limit is 0 . If denominator approaches zero faster, limit is \infty . If comparable, limit is a finite non-zero number. @keyframes run1 { 0% { left: 10%; } 50% { left: 85%; } 100% { left: 10%; } } @keyframes run2 { 0% { left: 10%; } 50% { left: 60%; } 100% { left: 10%; } }

Method 1: L'Hôpital's Rule

L'Hôpital's Rule Condition: Applicable ONLY if lim_{x \to a} f (x) = 0 and lim_{x \to a} g (x) = 0 (or both \infty). f (x) / g (x) \Rightarrow f^{'} (x) / g^{'} (x) Check if form is \frac{0}{0} or \frac{\infty}{\infty} . Differentiate numerator and denominator separately . Evaluate the new limit. If still indeterminate, repeat. Warning: Do NOT apply quotient rule (\frac{u}{v})^{'} . Differentiate u and v independently. @keyframes pulse-deriv { 0%, 100% { opacity: 1; transform: scale(1); } 50% { opacity: 0.7; transform: scale(1.05); } } .num-deriv, .den-deriv { animation: pulse-deriv 2s infinite ease-in-out; }

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Question 1

Level 1: Warm-up

Consider the standard limits L_{1} = lim_{x \to 0} \frac{e ^{x} - 1}{x}, L_{2} = lim_{x \to 0} \frac{l n ( 1 + x )}{x}, and L_{3} = lim_{x \to 0} \frac{e ^{2 x} - 1}{x} . What is the maximum value among L_{1}, L_{2}, L_{3} ?

Question 2

Level 1: Warm-up

To evaluate the following limit using L'Hôpital's rule: x \to 0 lim \frac{e ^{x} - x - 1}{x ^{2}} What is the minimum number of times the rule must be applied to resolve the indeterminate form?

Question 3

Level 1: Warm-up

What is the value of the following standard limit, which serves as a core engine for evaluating indeterminate forms? x \to 0 lim \frac{sin x}{x}

Question 4

Level 1: Warm-up

What is the value of the following standard limit, which serves as a fundamental building block introduced in the limits chapter? x \to 0 lim \frac{7 ^{x} - 1}{x}

Question 5

Level 1: Warm-up

A function f (x) outputs 1 if the first decimal digit d_{1} is even, and 0 if d_{1} is odd. What is the minimum number of points of discontinuity for f (x) in the open interval (0, 1) ?

Question 6

Level 1: Warm-up

If f (x) = x^{2} and g (x) = x^{3}, what is the value of the following limit? x \to 0^{-} lim \frac{f ( x )}{g ( x )}

Question 7

Level 1: Warm-up

If f (x) = x^{2} and g (x) = x^{5}, what is the value of the following limit? x \to 0^{-} lim \frac{f ( x )}{g ( x )}

Question 8

Level 1: Warm-up

Consider a piecewise function f (x) defined as g (x) for x < c and h (x) for x \geq c . Assuming g (x) and h (x) are continuous on their respective open intervals, which of the following statements is true?

Question 9

Level 1: Warm-up

If a function f (x) depends on the third decimal digit d_{3} of x, what is the maximum number of critical points (boundaries where d_{3} changes) in the open interval (0, 0.01) ?

Question 10

Level 1: Warm-up

In evaluating the limit lim_{x \to 0^{+}} \frac{x}{1 - e ^{3 x}} using the substitution t = x, which of the following statements is true regarding the bounds of t as x \to 0^{+} ?

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Limits and Continuity Notes for GATE CS

Limits and Continuity notes for GATE CS: 22 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Limits and Continuity

Chapter Journey: Limits & Continuity

Evaluation of Indeterminate Limits
L'Hôpital's Rule, Standard Limits, Series Expansion. The core engine.

Continuity of Piecewise Functions
Matching left-hand and right-hand limits. Finding unknown constants.

Discontinuities of Digit-Defined Functions
Advanced logic-based continuity checks. Rare but high-difficulty.

Goal: Master the algebraic manipulation of limits first. Everything else depends on it.

The Hero Concept: Why Indeterminate Forms?

What is an Indeterminate Form?

A limit $lim_{x \to a} \frac{f ( x )}{g ( x )}$ is indeterminate if direct substitution yields:

\frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, \infty - \infty, 1^{\infty}, 0^{0}, \infty^{0}

Numerator vs Denominator approaching zero

If numerator approaches zero faster, limit is $0$ .
If denominator approaches zero faster, limit is $\infty$ .
If comparable, limit is a finite non-zero number.

Method 1: L'Hôpital's Rule

L'Hôpital's Rule

Condition: Applicable ONLY if $lim_{x \to a} f (x) = 0$ and $lim_{x \to a} g (x) = 0$ (or both $\infty$ ).

f (x)

g (x)

\Rightarrow

f^{'} (x)

g^{'} (x)

Check if form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .
Differentiate numerator and denominator separately.
Evaluate the new limit.
If still indeterminate, repeat.

Warning: Do NOT apply quotient rule

(\frac{u}{v})^{'}

. Differentiate

u

and

v

independently.

Worked Example: Square Root Limit

Problem:

x \to 0^{+} lim \frac{x}{1 - e ^{2 x}}

Step 1: Substitution
Let

t = x

. As

x \to 0^{+}

t \to 0^{+}

L = t \to 0^{+} lim \frac{t}{1 - e ^{2 t}}

Step 2: L'Hôpital's Rule
Form is

\frac{0}{0}

. Differentiate top and bottom:

L = t \to 0^{+} lim \frac{1}{- 2 e ^{2 t}}

Step 3: Evaluate
Substitute

t = 0

L = \frac{1}{- 2 e ^{0}} = - 0.5

Answer:

- 0.5

Limits and Continuity: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Engineering Mathematics MCQ

Consider the standard limits $L_{1} = lim_{x \to 0} \frac{e ^{x} - 1}{x}$ , $L_{2} = lim_{x \to 0} \frac{l n ( 1 + x )}{x}$ , and $L_{3} = lim_{x \to 0} \frac{e ^{2 x} - 1}{x}$ . What is the maximum value among $L_{1}, L_{2}, L_{3}$ ?

A.
2
B.
1
C.
$ln 2$
D.
e

Correct Answer:

Step-by-Step Solution

Key idea: The standard limits evaluate directly to known constants.

Step 1: Recall the standard limit $lim_{x \to 0} \frac{e ^{x} - 1}{x} = 1$ . Thus, $L_{1} = 1$ .

Step 2: Recall the standard limit $lim_{x \to 0} \frac{l n ( 1 + x )}{x} = 1$ . Thus, $L_{2} = 1$ .

Step 3: Evaluate $L_{3}$ by adjusting the coefficient: $lim_{x \to 0} \frac{e ^{2 x} - 1}{x} = 2 \cdot lim_{x \to 0} \frac{e ^{2 x} - 1}{2 x} = 2 \cdot 1 = 2$ .

Step 4: Compare the values: $L_{1} = 1$ , $L_{2} = 1$ , $L_{3} = 2$ . The maximum value is 2.

Answer: 2

Question 2 · Engineering Mathematics MCQ

To evaluate the following limit using L'Hôpital's rule:

$x \to 0 lim \frac{e ^{x} - x - 1}{x ^{2}}$

What is the minimum number of times the rule must be applied to resolve the indeterminate form?

A.
1
B.
3
C.
2
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: Track the form after each application of L'Hôpital's rule.

Step 1: Check initial form. As $x \to 0$ , $e^{x} - x - 1 \to 0$ and $x^{2} \to 0$ . Form is $0/0$ .

Step 2: Apply L'Hôpital's rule (1st time). Derivative of top is $e^{x} - 1$ . Derivative of bottom is $2 x$ . New limit: $lim_{x \to 0} \frac{e ^{x} - 1}{2 x}$ .

Step 3: Check new form. As $x \to 0$ , $e^{x} - 1 \to 0$ and $2 x \to 0$ . Form is still $0/0$ .

Step 4: Apply L'Hôpital's rule (2nd time). Derivative of top is $e^{x}$ . Derivative of bottom is $2$ . New limit: $lim_{x \to 0} \frac{e ^{x}}{2}$ .

Step 5: Evaluate. $\frac{e ^{0}}{2} = \frac{1}{2}$ . The form is resolved.

Answer: 2

Question 3 · Engineering Mathematics MCQ

What is the value of the following standard limit, which serves as a core engine for evaluating indeterminate forms?

$x \to 0 lim \frac{sin x}{x}$

A.
0
B.
1
C.
$\infty$
D.
undefined

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall of a fundamental standard limit.

Step 1: Identify the form. As $x \to 0$ , $sin x \to 0$ and $x \to 0$ . This is a $0/0$ indeterminate form.

Step 2: Apply the standard limit formula $lim_{x \to 0} \frac{s i n x}{x} = 1$ .

Step 3: The value is exactly 1.

Answer: 1

Question 4 · Engineering Mathematics MCQ

What is the value of the following standard limit, which serves as a fundamental building block introduced in the limits chapter?

$x \to 0 lim \frac{7 ^{x} - 1}{x}$

A.
0
B.
1
C.
$ln 7$
D.
7

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall of a fundamental standard limit for exponential functions.

Step 1: Identify the form. As $x \to 0$ , $7^{x} - 1 \to 0$ and $x \to 0$ . This is a $0/0$ indeterminate form.

Step 2: Apply the standard limit formula $lim_{x \to 0} \frac{a ^{x} - 1}{x} = ln a$ .

Step 3: Substitute $a = 7$ into the formula. The value is exactly $ln 7$ .

Answer: $ln 7$

Question 5 · Engineering Mathematics MCQ

A function $f (x)$ outputs 1 if the first decimal digit $d_{1}$ is even, and 0 if $d_{1}$ is odd. What is the minimum number of points of discontinuity for $f (x)$ in the open interval $(0, 1)$ ?

A.
8
B.
9
C.
10
D.
100

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction/analysis problem to count discontinuities based on digit parity.

Step 1: Identify the target digit and boundaries. The function depends on $d_{1}$ , so boundaries are at $0.1, 0.2, \dots, 0.9$ .

Step 2: Check for jumps at each boundary. At any boundary $m /10$ , $d_{1}$ changes from $(m - 1) (mod 10)$ to $m (mod 10)$ . Since one is even and the other is odd, the function value always flips between 0 and 1.

Step 3: Count the discontinuities in $(0, 1)$ . Every boundary is a discontinuity. There are 9 boundaries ( $0.1$ to $0.9$ ).

Answer: 9

Question 6 · Engineering Mathematics MCQ

If $f (x) = x^{2}$ and $g (x) = x^{3}$ , what is the value of the following limit?

$x \to 0^{-} lim \frac{f ( x )}{g ( x )}$

A.
0
B.
1
C.
$\infty$
D.
$- \infty$

Correct Answer:

Step-by-Step Solution

Key idea: Compare the rate at which the numerator and denominator approach zero, and pay attention to the direction of the limit.

Step 1: Substitute the functions: $lim_{x \to 0^{-}} \frac{x ^{2}}{x ^{3}}$ .

Step 2: Simplify the expression for $x \neq = 0$ : $\frac{x ^{2}}{x ^{3}} = \frac{1}{x}$ .

Step 3: Evaluate the limit of the simplified expression as $x \to 0^{-}$ . Since $x$ is a small negative number, $\frac{1}{x}$ approaches $- \infty$ .

Answer: $- \infty$

Question 7 · Engineering Mathematics MCQ

If $f (x) = x^{2}$ and $g (x) = x^{5}$ , what is the value of the following limit?

$x \to 0^{-} lim \frac{f ( x )}{g ( x )}$

A.
0
B.
1
C.
$\infty$
D.
$- \infty$

Correct Answer:

Step-by-Step Solution

Key idea: Compare the rate at which the numerator and denominator approach zero, and pay attention to the direction of the limit.

Step 1: Substitute the functions: $lim_{x \to 0^{-}} \frac{x ^{2}}{x ^{5}}$ .

Step 2: Simplify the expression for $x \neq = 0$ : $\frac{x ^{2}}{x ^{5}} = \frac{1}{x ^{3}}$ .

Step 3: Evaluate the limit of the simplified expression as $x \to 0^{-}$ . Since $x$ is a small negative number, $x^{3}$ is a small negative number, so $\frac{1}{x ^{3}}$ approaches $- \infty$ .

Answer: $- \infty$

Question 8 · Engineering Mathematics MCQ

Consider a piecewise function $f (x)$ defined as $g (x)$ for $x < c$ and $h (x)$ for $x \geq c$ . Assuming $g (x)$ and $h (x)$ are continuous on their respective open intervals, which of the following statements is true?

A.
$f (x)$ is continuous everywhere if $lim_{x \to c^{-}} g (x) = lim_{x \to c^{+}} h (x)$ .
B.
$f (x)$ can only be discontinuous at the junction point $x = c$ .
C.
$f (x)$ is continuous at $x = c$ if $g (c) = h (c)$ .
D.
$f (x)$ is always discontinuous at $x = c$ because the rules change.

Correct Answer:

Step-by-Step Solution

Key idea: A piecewise function is only at risk of breaking at the boundaries where the rules change.

Step 1: Analyze the intervals. For $x < c$ , $f (x) = g (x)$ . Since $g (x)$ is continuous on its open interval, $f (x)$ is continuous for all $x < c$ .

Step 2: For $x > c$ , $f (x) = h (x)$ . Since $h (x)$ is continuous on its open interval, $f (x)$ is continuous for all $x > c$ .

Step 3: The only point where the rule changes is $x = c$ . Therefore, the only possible point of discontinuity is at the junction $x = c$ .

Answer: $f (x)$ can only be discontinuous at the junction point $x = c$ .

Question 9 · Engineering Mathematics MCQ

If a function $f (x)$ depends on the third decimal digit $d_{3}$ of $x$ , what is the maximum number of critical points (boundaries where $d_{3}$ changes) in the open interval $(0, 0.01)$ ?

A.
9
B.
10
C.
99
D.
100

Correct Answer:

Step-by-Step Solution

Key idea: This is an observation problem to count the boundaries for a specific decimal digit.

Step 1: Identify the target digit and step size. The function depends on $d_{3}$ , so boundaries occur at multiples of $1 0^{- 3} = 0.001$ .

Step 2: List the boundaries in the interval $(0, 0.01)$ . The multiples of $0.001$ strictly between $0$ and $0.01$ are $0.001, 0.002, \dots, 0.009$ .

Step 3: Count the points. There are exactly 9 such points.

Answer: 9

Question 10 · Engineering Mathematics MCQ

In evaluating the limit $lim_{x \to 0^{+}} \frac{x}{1 - e ^{3 x}}$ using the substitution $t = x$ , which of the following statements is true regarding the bounds of $t$ as $x \to 0^{+}$ ?

A.
$t \to 0^{+}$ only
B.
$t \to 0$ from both sides
C.
$t \to 0^{-}$ only
D.
$t$ can be any real number

Correct Answer:

Step-by-Step Solution

Key idea: The domain of the square root function restricts the substitution.

Step 1: The original limit specifies $x \to 0^{+}$ , meaning $x$ approaches 0 from the positive side ( $x > 0$ ).

Step 2: We substitute $t = x$ . The square root function $x$ is only defined for $x \geq 0$ , and its output is always non-negative ( $t \geq 0$ ).

Step 3: As $x \to 0^{+}$ , $t = x$ must also approach 0 from the positive side. Therefore, $t \to 0^{+}$ only.

Answer: $t \to 0^{+}$ only

limits and continuity notes

Chapter Roadmap: Limits and Continuity

Chapter Journey: Limits & Continuity

The Hero Concept: Why Indeterminate Forms?

What is an Indeterminate Form?

Method 1: L'Hôpital's Rule

L'Hôpital's Rule

Try a question

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Limits and Continuity Notes for GATE CS

Chapter Roadmap: Limits and Continuity

Chapter Journey: Limits & Continuity

The Hero Concept: Why Indeterminate Forms?

What is an Indeterminate Form?

Method 1: L'Hôpital's Rule

L'Hôpital's Rule

Worked Example: Square Root Limit

Worked Example: Square Root Limit

Limits and Continuity: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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