CBSE Class 11 - Mathematics - Limits and Derivatives (Part-3) Questions and Answers #class11Maths #Limits #eduvictors

CBSE Class 11 - Mathematics - Limits and Derivatives 

Part-3  Algebra of Limits  

#class11Maths #Limits 

In the previous post Limits and derivatives Part-2, we get basic ideas of the algebra of limits and also learned about rules and properties of limits.

Let us try to solve few problems:

Q1: Evaluate the given limit $\lim_{r\rightarrow 1} \pi r^2$

Answer: $\lim_{r\rightarrow 1} \pi r^2 = \pi(1)^2 = \pi$

Q2: Evaluate the given limit $\lim_{x\rightarrow 1} \frac{x^3-1}{x-1}$

Answer: 

$\lim_{x\rightarrow 1} \frac{x^3-1}{x-1}$

= $ \lim_{x\rightarrow 1} \frac{(x-1)(x^2 + x + 1)}{x-1}$

= $ \lim_{x\rightarrow 1} (x^2 + x + 1) $

= $1^2 + 1 + 1 = 3$

Q3: Evaluate the following limit:

$\lim_{x\rightarrow 2} \frac{x^3-3x+2}{x^2 + x - 6}$

Answer:

$\lim_{x\rightarrow 2} \frac{x^3-3x+2}{x^2 + x - 6}$

=$ \lim_{x\rightarrow 2} \frac{(x-2)(x-1)}{(x-2)(x+3)} $

=$\lim_{x\rightarrow 2} \frac{(x-1)}{(x+3)}$

=$ \frac{2-1}{2+3}$

=$\frac{1}{5}$

Let us prove another theorem

Theorem: For any positive integer n, (or nis any positive rational number)

$\lim_{x\rightarrow a} \frac{x^n-a^n}{x - a} =na^{n-1}$

Proof:

$\because x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + ... + xa^{n-2} + a^{n-1})$

$\lim_{x\rightarrow a} \frac{x^n-a^n}{x - a} =na^{n-1}$

= $ lim_{x\rightarrow a} \frac{(x - a)(x^{n-1} + x^{n-2}a + ... + xa^{n-2} + a^{n-1})}{x-a}$

= $lim_{x\rightarrow a} (x^{n-1} + x^{n-2}a + ... + xa^{n-2} + a^{n-1})$ 

= $a^{n-1} + a^{n-2}.a + a^{n-3}.a^2+ ... + a.a^{n-2} + a^{n-1}$

= $a^{n-1} + a^{n-1} + a^{n-1} + ... + a^{n-1} +  a^{n-1}$

= $n a^{n-1}$

Q4: Evaluate the given limit $\lim_{x\rightarrow 0}\frac{(x+1)^5 - 1}{x}$ 

Answer: Let x + 1 = y 

Since x→0, therefore y→1

We have, 

$\lim_{x\rightarrow 0}\frac{(x+1)^5 - 1}{x}$ 

$= \lim_{y\rightarrow 1}\frac{y^5 - 1}{y -1}$

Since, $\lim_{x\rightarrow a} \frac{x^n-a^n}{x - a} =na^{n-1}$

= $5.1^{5-1}$

= 5

Q5: Evaluate the given limit

$\lim_{z\rightarrow 1}\frac{z^\frac{1}{3} - 1}{z^\frac{1}{6} -1}$

Answer: At z = 1 the given function becomes indeterminate.

Let $z\frac{1}{6} = x$.

Since z→1, therefore x→1

= $ \lim_{z\rightarrow 1} \frac{x^2 -1}{x - 1}$

Since, $\lim_{x\rightarrow a} \frac{x^n-a^n}{x - a} =na^{n-1}$

= $2.1^{2-1}$

= 2

Q6: Evaluate $\lim_{x\rightarrow 2} \frac{x^{5} - 32}{x^3 - 8}$

Answer: 

$\lim_{x\rightarrow 2} \frac{x^{5} - 32}{x^3 - 8}$

= $\lim_{x\rightarrow 2} \frac{x^{5} - 2^5}{x^3 - 2^3}$

= $\frac{\lim_{x\rightarrow 2}(x^{5} - 2^5)}{\lim_{x\rightarrow 2}(x^3 - 2^3)}$

= $\frac{\lim_{x\rightarrow 2}\frac{(x^{5} - 2^5)}{x-2}}{\lim_{x\rightarrow 2}\frac{(x^3 - 2^3)}{x-2}}$

Since, $\lim_{x\rightarrow a} \frac{x^n-a^n}{x - a} =na^{n-1}$

=$\frac{5\times2^4}{3 \times2^2}$

=$\frac{80}{12}$

=$\frac{20}{3}$

Q7: Evaluate $\lim_{x\rightarrow a}\frac{(x^{m} - a^m)}{(x^{n} - a^n)}$

Answer:

$ \lim_{x\rightarrow a}\frac{(x^{m} - a^m)}{(x^{n} - a^n)}$

= $\lim_{x\rightarrow a}\frac{(x^{m} - a^m)}{(x^{n} - a^n)}$

= $\frac{\lim_{x\rightarrow a}(x^m - a^m)}{\lim_{x\rightarrow a}(x^n - a^n)}$

= $\frac{\lim_{x\rightarrow a}\frac{(x^m - a^m)}{x-a}}{\lim_{x\rightarrow a}\frac{(x^n - a^n)}{x-a}}$

Since, $\lim_{x\rightarrow a} \frac{x^n-a^n}{x - a} =na^{n-1}$

=$\frac{m.a^{m-1}}{n. a^{n-1}}$

=$\frac{m}{n}a^{m-n}$

👉Q6 can be solved directly by using the above-said formula.

In the next post, we'll learn about trigonometry limits.

👉See Also:

Ch13 Limits and Derivatives - Q & A (Part-1)

Ch 13 Limits and Derivatives - Q & A (Part - 2)

SETS (NCERT Ex 1.1)
SETS (NCERT Ex 1.2)
Different Types of Sets
SETS (NCERT Ex 1.3)

SETS (Unit Test Paper)
SETS (VENN DIAGRAMS)
SETS (Operations of Sets)
SETS (NCERT Ex 1.4 Q1-Q5)
SETS (NCERT Ex 1.4 Q6 - Q8)
SETS (NCERT Ex 1.4 Q9 - Q12)
SETS (NCERT Ex 1.5)
SETS (NCERT Ex 1.6) 
Laws of Set Operations
Ch2: Relations and Functions (1 Mark Q & A) Part-1
Ch2: Cartesian Product of Two Sets (Important Points)
Ch2: Relations - Domain, Range and Co-Domain (Solved Problems)

Ch5: Complex Numbers (Part 1) - Solved Problems

Maths Annual Test Paper (2018-19)
Maths Annual Test Paper (2020-21) 
Maths Term1 MCQs (2021-22)