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📝 M delta definition of infinite limits (15 MCQs)

📖 From Calculus • 2. Limits and Continuity an Introduction • 15 questions available

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What is M delta definition of infinite limits?

Definition:
The M-delta definition for an infinite limit $\lim_{x \to a} f(x) = \infty$ states that for every positive number $M$ , there exists a $\delta > 0$ such that if $0 < |x-a| < \delta$ , then $f(x) > M$ . This formalizes the idea that $f(x)$ can be made arbitrarily large by taking $x$ sufficiently close to $a$ , providing a rigorous basis for vertical asymptotes and unbounded behavior.

Example:
Prove $\lim_{x \to 1} \frac{1}{(x-1)^2} = \infty$ .
Solution: For $M > 0$ , choose $\delta = 1/\sqrt{M}$ . If $0 < |x-1| < \delta$ , then $1/(x-1)^2 > 1/\delta^2 = M$ .

Reason:
This definition is essential for rigorously proving infinite limits, which is necessary for understanding singularities in physical models and for advanced calculus topics like improper integrals.

Easy

Medium

Hard

📝 All M delta definition of infinite limits MCQs

Q1. According to Definition 1.4.2, what must exist for every  $\varepsilon>0$ when $\lim_{x\to +\infty}f(x)=L$ ?

A.A positive number 

N

such that

|f(x)-L|<\varepsilon

whenever

x>N

✅

B.A negative number 

N

such that

|f(x)-L|<\varepsilon

whenever

x<N

C.A number 

M

such that

|f(x)-L|<\varepsilon

whenever

|x|>M

D.A CONSTANT 

C

WITH

|F(C)-L|<\VAREPSILON

💡 Difficulty: easy | ✅ Correct: A

Q2. For the linear function $f(x)=2x+3$ , what is $\displaystyle\lim_{x\to +\infty}f(x)$ ?

2

+\infty

✅

C.Does not exist

3

💡 Difficulty: easy | ✅ Correct: B

Q3. If $f(x)=5$ for all real  $x$ , what is $\displaystyle\lim_{x\to +\infty}f(x)$ ?

0

+\infty

5

✅

D.UNDEFINED

💡 Difficulty: easy | ✅ Correct: C

Q4. Which description correctly captures the behavior of $f(x)=\frac{1}{x}$ as $x\to +\infty$ ?

A.Approaches 

1

B.Diverges to 

+\infty

C.Oscillates between 

-1

and 

1

D.APPROACHES 

0

✅

💡 Difficulty: easy | ✅ Correct: D

Q5. In the $\varepsilon$ – $N$ definition, if $\varepsilon=0.01$ for $f(x)=\frac{1}{x}$ , which of the following could serve as a valid  $N$ ?

100

✅

0.01

-100

1

💡 Difficulty: easy | ✅ Correct: A

Q6. Suppose $\lim_{x\to +\infty}f(x)=3$ and $\lim_{x\to +\infty}g(x)=-2$ . What is $\displaystyle\lim_{x\to +\infty}(f(x)+g(x))$ ?

5

1

✅

-5

D.DOES NOT EXIST

💡 Difficulty: medium | ✅ Correct: B

Q7. Compare the limits of $h(x)=\frac{1}{x}$ and $k(x)=\frac{1}{x^{2}}$ as $x\to +\infty$ . Which statement is true?

A.Both limits are 

0

and

h(x)

approaches 

0

faster than

k(x)

B.Both limits are 

0

and

k(x)

approaches 

0

faster than

h(x)

C.Both limits are 

0

and

k(x)

approaches 

0

faster than

h(x)

✅

D.NONE OF THE ABOVE

💡 Difficulty: medium | ✅ Correct: C

Q8. Which of the following $\varepsilon$ – $N$ statements correctly captures $\displaystyle\lim_{x\to +\infty}f(x)=L$ for $f(x)=3+\frac{1}{x}$ ?

A.For every

\varepsilon>0

there exists

N

such that if

x<N

then

|f(x)-L|<\varepsilon

B.For some

\varepsilon>0

there exists

N

such that if

x>N

then

|f(x)-L|<\varepsilon

C.If

|f(x)-L|<\varepsilon

then

x>N

for some

N

D.FOR EVERY

\VAREPSILON>0

THERE EXISTS

N

SUCH THAT IF

X>N

THEN

|F(X)-L|<\VAREPSILON

✅

💡 Difficulty: medium | ✅ Correct: D

Q9. For $f(x)=\sin x$ , which statement about $\displaystyle\lim_{x\to +\infty}f(x)$ is correct?

A.The limit does not exist ✅

B.The limit equals 

0

C.The limit equals 

1

D.THE LIMIT EQUALS 

-1

💡 Difficulty: medium | ✅ Correct: A

Q10. If a function satisfies: for every $\varepsilon>0$ there exists $N$ such that $x>N\Rightarrow|f(x)-L|<\varepsilon$ , which of the following must be false?

f

is unbounded as

x\to +\infty

✅

f

approaches

L

x\to +\infty

f

is eventually within

\varepsilon

L

N

CAN BE CHOSEN DEPENDENT ON

\VAREPSILON

💡 Difficulty: medium | ✅ Correct: A

Q11. Given $f(x)=\frac{x}{x+1}$ , which $N$ guarantees $|f(x)-1|<0.1$ for all $x>N$ ?

N=1

N=0

N=9

✅

N=100

💡 Difficulty: medium | ✅ Correct: C

Q12. Which function does NOT have a finite limit as $x\to +\infty$ ?

\displaystyle\frac{1}{x}

5

e^{-x}

X^{2}

✅

💡 Difficulty: medium | ✅ Correct: D

Q13. To prove $\displaystyle\lim_{x\to +\infty}\frac{x^{2}}{x+1}=+\infty$ using the definition, which intermediate statement is correct?

A.For any

M>0

choose

N

such that

x>N\Rightarrow\frac{x^{2}}{x+1}>M

✅

B.For any

M>0

choose

N

such that

x<N\Rightarrow\frac{x^{2}}{x+1}>M

C.For any

M>0

choose

N

such that

x>N\Rightarrow\frac{x^{2}}{x+1}<M

D.FOR ANY

M>0

CHOOSE

N

SUCH THAT

X<N\RIGHTARROW\FRAC{X^{2}}{X+1}<M

💡 Difficulty: hard | ✅ Correct: A

Q14. If $\lim_{x\to +\infty}f(x)=L$ and $\lim_{x\to +\infty}g(x)=L$ , must $\displaystyle\lim_{x\to +\infty}(f(x)\cdot g(x))=L^{2}$ ?

A.No, it can fail for some functions

B.Yes, always ✅

C.Only if both functions are bounded

D.CANNOT BE DETERMINED WITHOUT MORE INFORMATION

💡 Difficulty: hard | ✅ Correct: B

Q15. Consider the sequence $a_{n}=n\sin\!\left(\frac{1}{n}\right)$ . Using the $\varepsilon$ – $N$ definition for limits as $x\to +\infty$ , what is $\displaystyle\lim_{n\to\infty}a_{n}$ ?

0

B.Does not exist

1

✅

+\INFTY

💡 Difficulty: hard | ✅ Correct: C

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