Engineering Mathematics Multiple Choice Questions 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
Answer: c
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) ∞
Answer: c
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
Answer: c
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
Answer: b
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
Answer: d
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
Answer: a
Explanation: Converting into Polar form we have
=(lt_{rrightarrow 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_{rrightarrow 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_{rrightarrow 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
Answer: d
Explanation: Put x = t : y = at
(=lt_{trightarrow 0}frac{sin(at)}{t})
(=lt_{trightarrow 0}a times frac{sin(at)}{at}=a times lt_{trightarrow 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
Answer: d
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_{trightarrow 0}frac{sin(t)}{at}=frac{1}{a} lt_{trightarrow 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) ∞
Answer: b
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
Answer: d
Explanation: Put x = at : y = t
(=lt_{trightarrow 0}frac{sin(t)}{a^n.t^n})
(=lt_{trightarrow 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
Answer: a
Explanation: Put x = at : y = t
(=lt_{trightarrow 0}frac{1}{a^nt^{n-1}}times frac{sin(sin(t))}{t})
(=lt_{trightarrow 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
Answer: d
Explanation: Put x = t : y = at
=(lt_{trightarrow 0}times frac{tan(at)}{t})
=(lt_{trightarrow 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
Answer: a
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
Answer: a
Explanation: lt(x, y)→(0, 0)sinh(x)⁄x * lt(x, y)→(0, 0)sinh(y)⁄y
= 1 * 1
= 1.