Find the limit and discuss the continuity of the function. lim_{(x,y)rightarrow(3,1)}(x^2-2y)

Question
Limits and continuity
asked 2021-01-04
Find the limit and discuss the continuity of the function. \(\lim_{(x,y)\rightarrow(3,1)}(x^2-2y)\)

Answers (1)

2021-01-05
Rewiew the given function
\(f(x,y)=(x^2-2y)\)
Objective is to find the limit when \((x,y)\Rightarrow(3,1)\) and discuss the continuity of the function.
To find the limit when \((x,y)\Rightarrow(3,1)\), solve
\(=\lim_{(x,y)\rightarrow(3,1)}(x^2-2y)\)
Since the above limit is not forming any indeterminate form, so evaluate the just by substituting the value of x and y.
\(=(3^2-2\cdot1)\)
\(=9-2\)
\(=7\)
To check the continuity of the function
Any function f(x,y) is a continuous point \((x_0,y_0)\) if \(\lim_{(x,y)\rightarrow(x_0,y_0)}f(x,y)=f(x_0,y_0)\)
Here
\(\lim_{(x,y)\rightarrow(x_0,y_0)}f(x,y)=f(x_0,y_0)=\lim_{(x,y)\rightarrow(3,1)}x^2-2y=7\) and \(f(x_0,y_0)=f(3,1)=3^2-2\cdot1=7\)
Since,
\(\lim_{(x,y)\rightarrow(x_0,y_0)} f(x,y)=f(x_0,y_0)=7\)
So, the given function is continuous at point (3,1).
0

Relevant Questions

asked 2020-11-27
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(0,\pi/2)}\frac{1-\cos xy}{4x^2y^3}\)
asked 2020-11-01
Evaluate the following limits.
\(\lim_{(x,y)\rightarrow(2,-1)}(xy^8-3x^2y^3)\)
asked 2021-02-26
Find the function The following limits represent the slope of a curve y=f(x) at the point {a,f(a)} Determine a possible function f and number a, then calculate the limit
\(\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}\)
asked 2021-02-21
Find the following limit:
\(\lim_{(x,y)\rightarrow(3,3)}(\frac{x-y}{\sqrt x-\sqrt y})\)
asked 2021-01-10
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(1,1)}\frac{x^2+xy-2y^2}{2x^2-xy-y^2}\)
asked 2021-03-07
Find the limits:
\(\lim_{(x,y)\rightarrow(0,0)}\cos\frac{x^2+y^3}{x+y+1}\)
asked 2020-10-28
Find the limits
\(\lim_{(x,y)\rightarrow(3,4)}\sqrt{x^2+y^2-1}\)
asked 2021-02-03
A piecewise function is given. Use properties of limits to find the shown indicated limit, or state that the limit does not exist. \(f(x)=\begin{cases}\sqrt[3]{x^2+7}&\text{if}&x<1\\4x&\text{if}&x\geq1\end{cases}\)
\(\lim_{x\rightarrow1} f(x)\)
asked 2021-03-09
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
\(\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}\)
asked 2021-03-06
Use the method of your choice to evaluate the following limits.
\(\lim_{(x,y)\rightarrow(2,0)}\frac{1-\cos y}{xy^2}\)
...