This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Limits and Derivatives of Several Variables – 1”.
1. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{121.x^{-5}.y^{\frac{13}{3}}}{y+(x)\frac{3}{2}}\)
a) ∞
b) 0
c) Does Not Exist
d) 121
View Answer
Explanation: Put x = t : y = a.t3⁄2 we have
=\(lt_{(x,y)\rightarrow(0,0)}\frac{121.t^{-5}.(at^{\frac{3}{2}})^{\frac{13}{3}}}{t^{\frac{3}{2}}+t^{\frac{3}{2}}}\)
=\(lt_{(x,y)\rightarrow(0,0)}\frac{121.at^{\frac{13}{3}}.t^{\frac{3}{2}}}{2.t^{\frac{3}{2}}}\)
=\(lt_{(x,y)\rightarrow(0,0)}\frac{121.at^{\frac{13}{3}}}{2}\)
By varying a we get different limits along different paths
Hence, Does Not Exist is the right answer.
2. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{y^6}{x^{10}y^2+x^{15}}\)
a) 0
b) 1
c) Does Not exist
d) ∞
View Answer
Explanation: Put Put x = t : y = a.t5⁄2 we have
=\(lt_{(x,y)\rightarrow(0,0)}\frac{(a.t^{\frac{5}{2}})^6}{t^{10}.(a.t^{\frac{5}{2}})^2+t^{15}}\)
=\(lt_{(x,y)\rightarrow(0,0)}\frac{a^6}{a^2+1}\)
By varying a we get different limits along different paths
Hence, Does Not exist is the right answer.
3. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sec(y).sin(x)}{x}\)
a) ∞
b) 1⁄2
c) 1
d) 1⁄3
View Answer
Explanation: Treating limits separately we have
lt(x, y)→(0, 0) sin(x)⁄x * lt(x, y)→(0,0) sec(y)
= 1 * 1
= 1.
4. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{x^3-y^3}{(x-y)}\)
a) -1⁄2
b) 0
c) ∞
d) -90
View Answer
Explanation: Simplifying the expression we have
\(lt_{(x,y)\rightarrow(0,0)}\frac{(x-y)(x^2+xy+y^2)}{(x-y)}\)
\(lt_{(x,y)\rightarrow(0,0)}\frac{(x^2+xy+y^2)}{1}\)=(02+0.0+02)
= 0.
5. Find \(lt_{(x,y)\rightarrow(0,1)}\frac{x+y-1}{\sqrt{x+y}-1}\)
a) 9
b) 0
c) 6
d) 2
View Answer
Explanation: Simplifying the expression we have
=\(lt_{(x,y)\rightarrow(0,1)}\frac{(\sqrt{x+y}^2)-(1)^2}{\sqrt{x+y}-1}=lt_{(x,y)\rightarrow(0,1)}\frac{(\sqrt{x+y}+1).(\sqrt{x+y}-1)}{\sqrt{x+y}-1}\)
=\(lt_{(x,y)\rightarrow(0,1)}(\sqrt{x+y}+1)=\sqrt{1}+1\)
=2
6. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{x^3+3xy^2-xy^2}{x^2+xy}\)
a) 0
b) ∞
c) 1
d) -1
View Answer
Explanation: Converting into Polar form we have
=\(lt_{r\rightarrow 0}\frac{r^3.cos^3(\theta))+3(r^2.cos^2(\theta))(r.sin(\theta))-(r.cos(\theta))(r^2.sin^2(\theta))}{(r^2cos^2(\theta))+r^2sin(\theta)cos(\theta)}\)
=\(lt_{r\rightarrow 0}\frac{r^3}{r^2}\times (\frac{cos^3(\theta)+3(cos^2(\theta))(sin(\theta))-(cos(\theta))(sin^2(\theta))}{(cos^2(\theta))+sin(\theta)cos(\theta)})\)
=\(lt_{r\rightarrow 0}(r)\times (\frac{cos^3(\theta)+3(cos^2(\theta))(sin(\theta))-(cos(\theta))(sin^2(\theta))}{(cos^2(\theta))+sin(\theta)cos(\theta)})\)
=0
7. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sin(y)}{x}\)
a) 1
b) 0
c) ∞
d) Does Not Exist
View Answer
Explanation: Put x = t : y = at
\(=lt_{t\rightarrow 0}\frac{sin(at)}{t}\)
\(=lt_{t\rightarrow 0}a \times \frac{sin(at)}{at}=a \times lt_{t\rightarrow 0}\frac{sin(at)}{at}\)
= a * (1) = a
By varying a we get different limits
Hence, Does Not Exist is the right answer.
8. Find \(lt_{(x,y)\rightarrow(\infty,0)}(\sum_{a=1}^{x-1}sin(\frac{a}{x}).sin(y))\)
a) 1
b) -1
c) ∞
d) Does not Exist
View Answer
Explanation: Multiplying and dividing by we have
\(lt_{(x,y)\rightarrow(\infty,0)}(sin(y))\times(\sum_{a=1}^{x-1}sin(\frac{a}{x}))\)
\(lt_{(x,y)\rightarrow(\infty,0)}(x.sin(y))\times lt_{(x,y)\rightarrow(\infty,0)}\left (\sum_{a=1}^{x-1}\frac{sin(\frac{a}{x})}{x}\right )\)
\(lt_{(x,y)\rightarrow(\infty,0)}(\frac{sin(y)}{\frac{1}{x}})\times lt_{(x,y)\rightarrow(\infty,0)}\left (\sum_{a=1}^{x-1}\frac{sin(\frac{a}{x})}{x}\right )\)
Put z=1/x : as x → ∞ : z → 0
Consider one part of the limit
\(=lt_{(x,y)\rightarrow (0,0)}\frac{sin(y)}{z}\)
Put : y = t : z = at
\(=lt_{t\rightarrow 0}\frac{sin(t)}{at}=\frac{1}{a} lt_{t\rightarrow 0}\frac{sin(t)}{t}\)
=\(\frac{1}{a}\times 1= \frac{1}{a}\).
9. Find \(lt_{(x,y)\rightarrow (0,0)}\frac{y^7x^{98}-x^{97}y^8+x^{105}}{xy^7+x^8}\)
a) Does Not Exist
b) 0
c) 1
d) ∞
View Answer
Explanation: Put x =r.cos(ϴ) : y = r.sin(ϴ)
=\(lt_{(x,y)\rightarrow (0,0)}\frac{(r^7.sin^7(\theta))(r^{98}.sin^{98}(\theta))-(r^{97}.cos^{97}(\theta))(r^8.sin^8(\theta))+(r^{105}.cos^{105}(\theta))}{(r.cos(\theta)(r^7.sin^7(\theta))+(r^8.cos(\theta))}\)
=\(lt_{(x,y)\rightarrow (0,0)}\frac{r^{105}}{r^8}\times \frac{(sin^7(\theta))(sin^{98}(\theta))-(cos^{97}(\theta))(sin^8(\theta))+(cos^{105}(\theta))}{(cos(\theta)(sin^7(\theta))+(cos(\theta))}\)
=\(lt_{(x,y)\rightarrow (0,0)}(r^{97})\times \frac{(sin^7(\theta))(sin^{98}(\theta))-(cos^{97}(\theta))(sin^8(\theta))+(cos^{105}(\theta))}{(cos(\theta)(sin^7(\theta))+(cos(\theta))}\)
= 0
10. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sin(y)}{x^n}\)
a) 0
b) ∞
c) 1
d) Does Not Exist
View Answer
Explanation: Put x = at : y = t
\(=lt_{t\rightarrow 0}\frac{sin(t)}{a^n.t^n}\)
\(=lt_{t\rightarrow 0}\frac{1}{a^nt^{n-1}}\frac{sin(t)}{t}\)
By varying n we get different limits
Hence, Does Not Exist is the right answer.
11. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sin(sin(y))}{x^n}\)
a) Does not Exist
b) 0
c) ∞
d) 1
View Answer
Explanation: Put x = at : y = t
\(=lt_{t\rightarrow 0}\frac{1}{a^nt^{n-1}}\times \frac{sin(sin(t))}{t}\)
\(=lt_{t\rightarrow 0}\frac{1}{a^nt^{n-1}} \times (1)\)
By varying n we get different values of limits.
12. Find \(=lt_{(x,y)\rightarrow (0,0)}\frac{tan(y)}{x}\)
a) ∞
b) 1
c) 1⁄2
d) Does Not Exist
View Answer
Explanation: Put x = t : y = at
=\(lt_{t\rightarrow 0}\times \frac{tan(at)}{t}\)
=\(lt_{t\rightarrow 0} (a) \times \frac{tan(at)}{at}\)
=a
By varying the value of a we get different limits.
13. Find \(lt_{(x,y,z)\rightarrow(0,0,0)}\frac{sinh(x)\times sinh(y)\times sinh(z)}{xyz}\)
a) 1
b) ∞
c) 0
d) 990
View Answer
Explanation:
\(=lt_{(x,y,z)\rightarrow(0,0,0)}\frac{sinh(x)}{x}\times lt_{(x,y,z)\rightarrow(0,0,0)}\frac{sinh(y)}{y}\times lt_{(x,y,z)\rightarrow(0,0,0)}\frac{sinh(z)}{z}\)
= 1 * 1 * 1
= 1.
14. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sinh(x)\times sinh(y)}{xy}\)
a) 1
b) ∞
c) 0
d) 990
View Answer
Explanation: lt(x, y)→(0, 0) sinh(x)⁄x * lt(x, y)→(0, 0) sinh(y)⁄y
= 1 * 1
= 1.
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