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

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:
\(\lim _{x \rightarrow 0}\) \(\frac { 1-cosx }{ x^2 }\)

Question 2.
\(\lim _{x \rightarrow ∞}\) \(\frac { 2x^2-3 }{ x^2-5x+3 }\)
Solution:

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

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:

Question 7.
\(\lim _{x \rightarrow 1}\) (\(\frac { 2 }{ x^2-1 }\) – \(\frac { x }{ x-1 }\))
Solution:

Question 8.
\(\lim _{x \rightarrow 0^+}\) x^x
Solution:
\(\lim _{x \rightarrow 0^+}\) x^x [0° indeterminate form
Let g(x) = x^x
Taking log on both sides
log g(x) = log x^x
log g(x) = x log x
\(\lim _{x \rightarrow 0^+}\) log g(x) = \(\lim _{x \rightarrow 0^+}\) x log x [0 × ∞ indeterminate form

Question 9.
\(\lim _{x \rightarrow ∞}\) (1 + \(\frac { 1 }{ x }\)) ^x
Solution:

Applying L’ Hôpital’s Rule

Exponentiating we get, \(\lim _{x \rightarrow ∞}\) g(x) = e¹ = e

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,

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 = A₀e^rt.
Solution:

Applying L-Hospital’s Rule

Hence Proved.

Read More:

Coforge