Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 7 Applications of Differential Calculus Ex 7.5 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 7 Applications of Differential Calculus Ex 7.5

Question 1.
Evaluate the following limits, if necessary use L’ Hôpital’s Rule:
\(\lim _{x \rightarrow 0}\) \(\frac { 1-cosx }{ x^2 }\)
Solution:
\(\lim _{x \rightarrow 0}\) \(\frac { 1-cosx }{ x^2 }\)
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 1

Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5

Question 2.
\(\lim _{x \rightarrow ∞}\) \(\frac { 2x^2-3 }{ x^2-5x+3 }\)
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 2

Question 3.
\(\lim _{x \rightarrow ∞}\) \(\frac { x }{ log x }\)
Solution:
\(\lim _{x \rightarrow ∞}\) \(\frac { x }{ log x }\) [ \(\frac { ∞ }{ ∞ }\) indeterminate form
Applying L’ Hôpital’s Rule
\(\lim _{x \rightarrow ∞}\) \(\frac { 1 }{ \frac{1}{x} }\) = \(\lim _{x \rightarrow ∞}\) x = ∞

Question 4.
\(\lim _{x \rightarrow \frac{π}{2}}\) \(\frac { secx }{ tanx }\)
Solution:
\(\lim _{x \rightarrow \frac{π}{2}}\) \(\frac { secx }{ tanx }\) [ \(\frac { ∞ }{ ∞ }\) indeterminate form
Simplifying, we get
\(\lim _{x \rightarrow \frac{π}{2}}\) \(\frac { 1 }{ sinx }\) = \(\frac { 1 }{ sin \frac{π}{2} }\) = 1

Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5

Question 5.
\(\lim _{x \rightarrow ∞}\) e-x√x
Solution:
\(\lim _{x \rightarrow ∞}\) e-x√x [0 × ∞ indeterminate form
The other form is \(\lim _{x \rightarrow ∞}\) \(\frac { √x }{ e^x }\)
[0 × ∞ indeterminate form
Applying L’ Hôpital’s Rule
= \(\lim _{x \rightarrow ∞}\) \(\frac { 1 }{ 2 \sqrt{xe^x} }\)
= 0

Question 6.
\(\lim _{x \rightarrow ∞}\) (\(\frac { 1 }{ sinx }\) – \(\frac { 1 }{ x }\))
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 3

Question 7.
\(\lim _{x \rightarrow 1}\) (\(\frac { 2 }{ x^2-1 }\) – \(\frac { x }{ x-1 }\))
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 4

Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5

Question 8.
\(\lim _{x \rightarrow 0^+}\) xx
Solution:
\(\lim _{x \rightarrow 0^+}\) xx [0° indeterminate form
Let g(x) = xx
Taking log on both sides
log g(x) = log xx
log g(x) = x log x
\(\lim _{x \rightarrow 0^+}\) log g(x) = \(\lim _{x \rightarrow 0^+}\) x log x [0 × ∞ indeterminate form
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 5

Question 9.
\(\lim _{x \rightarrow ∞}\) (1 + \(\frac { 1 }{ x }\)) x
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 6
Applying L’ Hôpital’s Rule
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 7
Exponentiating we get, \(\lim _{x \rightarrow ∞}\) g(x) = e1 = e

Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5

Question 10.
\(\lim _{x \rightarrow \frac{π}{2}}\) (sin x) tan x
Solution:
\(\lim _{x \rightarrow \frac{π}{2}}\) (sin x)tan x [1 indeterminate form]
Let g(x) = (sin x) tan x
Taking log on both sides,
log g(x) = tan x log sin x
\(\lim _{x \rightarrow \frac{π}{2}}\) log g(x) = \(\lim _{x \rightarrow \frac{π}{2}}\) \(\frac { log sin x }{ cot x }\)
[ \(\frac { 0 }{ 0 }\) Indeterminate form
Applying L’ Hôpital’s Rule
= \(\lim _{x \rightarrow \frac{π}{2}}\) (\(\frac { cotx }{ -cosec^2x }\)) = -1
exponentiating, we get
\(\lim _{x \rightarrow \frac{π}{2}}\) g(x) = e-1 = \(\frac { 1 }{ e }\)

Question 11.
\(\lim _{x \rightarrow 0^+}\) (cos x) \(\frac { 1 }{ x^2 }\)
Solution:
\(\lim _{x \rightarrow 0^+}\) (cos x) \(\frac { 1 }{ x^2 }\) [1 indeterminate form
let g(x) = (cos x)\(\frac { 1 }{ x^2 }\)
Taking log on both sides,
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 8

Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5

Question 12.
If an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A = A0(1 + \(\frac { r }{ n }\))nt. If the interest is compounded continuously, (that is as n → ∞), show that the amount after t years is A = A0ert.
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 9
Applying L-Hospital’s Rule
Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5 10
Hence Proved.

Samacheer Kalvi 12th Maths Guide Chapter 7 Applications of Differential Calculus Ex 7.5

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