1. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{x^3+3xy^2-xy^2}{x^2+xy}\)
a) 0
b) ∞
c) 1
d) -1
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
2. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sin(y)}{x}\)
a) 1
b) 0
c) ∞
d) Does Not Exist
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.
3. 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
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
\(=lt_{t\rightarrow 0}\frac{sin(t)}{at}=\frac{1}{a} lt_{t\rightarrow 0}\frac{sin(t)}{t}\)
=\(\frac{1}{a}\times 1= \frac{1}{a}\).
4. 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) ∞
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
5. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sin(y)}{x^n}\)
a) 0
b) ∞
c) 1
d) Does Not Exist
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.
6. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sin(sin(y))}{x^n}\)
a) Does not Exist
b) 0
c) ∞
d) 1
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.
7. Find \(=lt_{(x,y)\rightarrow (0,0)}\frac{tan(y)}{x}\)
a) ∞
b) 1
c) 1⁄2
d) Does Not Exist
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.
8. 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
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.
9. Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sinh(x)\times sinh(y)}{xy}\)
a) 1
b) ∞
c) 0
d) 990
Explanation: lt(x, y)→(0, 0) sinh(x)⁄x * lt(x, y)→(0, 0) sinh(y)⁄y
= 1 * 1
= 1.
10. Two men on a surface want to meet each other. They have taken the point (0, 0) as meeting point. The surface is 3-D and its equation is f(x,y) = \(\frac{x^{\frac{-23}{4}}y^9}{x+(y)^{\frac{4}{3}}}\). Given that they both play this game infinite number of times with their starting point as (908, 908) and (90, 180)
(choosing a different path every time they play the game). Will they always meet?
a) They will not meet every time
b) They will meet every time
c) Insufficient information
d) They meet with probability 1⁄2
Explanation: The question is asking us to simply find the limit of the given function exists as the pair (x, y) tends to (0, 0) (The two men meet along different paths taken or not)
Thus, put x = t : y = a(t)3⁄4
\(=lt_{(x,y)\rightarrow(0,0)}=lt_{t\rightarrow 0}\frac{a^9.t^{\frac{27}{4}}.t^{\frac{-23}{4}}}{t+a^{\frac{4}{3}}.t^{\frac{1}{1}}}\)
\(=lt_{t\rightarrow 0}\frac{t}{t} \times \frac{a^9}{1+a^{\frac{4}{3}}}\)
\(=lt_{t\rightarrow 0} \frac{a^9}{1+a^{\frac{4}{3}}}\)
By putting different values of a we get different limits
Thus, there are many paths that do not go to the same place.
Hence, They will not meet every time is the right answer.