Limit and continuity

1. PRESENTATION FOR
CALCULUS
TOPIC :- LIMIT AND CONTINUTY OF
THE FUNCTION OF SEVERAL
VARIABLES

2. LIMIT OF FUNCTION
• We say that a function f(x , y) approaches the limit
L as (x , y) approaches (xₒ , yₒ) ,and written as,
• Lim f(x , y) = L
(x , y)→(x₀ , y₀)
• For every ε>0,there exists δ>0 such that, |f(x,y)-
L|<ε,whenever
d((x , y),(x₀ , y₀))< δ

3. EXAMPLES
• 1.]
lim [(xy)/(x²+y²)]
(x , y) →(0 , 0)
→ lim
(x , y) →(0 , 0) [(xy)/(x²+y²)]
y=0
=0

4. → lim
(x , y) →(0 , 0) [(xy)/(x²+y²)]
x=0
=0
→ lim
(x , y) →(0 , 0) [(xy)/(x²+y²)]
y=x
= x²/2x²
=1/2

5. → lim (*D.N.E.=Does not exist)
(x , y) →(0 , 0) [(xy)/(x²+y²)]
y=-x
= -x²/2x²
=-1/2
Hence ,
lim
(x , y) →(0 , 0) [(xy)/(x²+y²)] ≠
y=x
lim
(x , y) →(0 , 0) [(xy)/(x²+y²)]
y=-x
So limit D.N.E.

6. • 2.]
lim [(x²y)/(x⁴+y²)]
(x , y) →(0 , 0)
→y=mx²
= lim
(x , y) →(0 , 0) [x²(mx²)/x⁴+m² x⁴]
y=mx²
= m/1+m²
• Which is independent from x & y.
• So the limit D.N.E.

7. • 3.]
lim [(x²-xy)/(√x-√y)]
(x , y) →(0 , 0)
= lim [x(x-y)/(√x-√y)]
(x , y) →(0 , 0)
= lim [x(√x-√y) (√x+√y)]/(√x-√y)
(x , y) →(0 , 0)
= lim x(√x+√y)
(x , y) →(0 , 0)
=0
• So the limit exists and its value is 0.

8. • 4.]
lim [(x³-y³)/(x²+y²)]
(x , y) →(0 , 0)
→ Taking y = mxⁿ
= lim
(x , y) →(0 , 0) (x³-m³x³ⁿ)/(x²+m²x²ⁿ)
y=mxⁿ
= lim x[(1-m³x³ⁿ̄̄̄̄ˉ³)/(1+m²x²ⁿ̄̄̄̄ˉ²)]
(x , y) →(0 , 0)

9. = 0
• So the limit exists and its value is 0.
5.] Find :- lim (x⁴ + 4x³y - 5xy²)
(x , y) →(5 , -2)
→ lim (x⁴ + 4x³y - 5xy²)
(x , y) →(5 , -2)
= (5)⁴ + 4(5)(-2)³ - 5(5)(-2)²
=-475

10. CONTINUTY OF A FUNCTION
• The function u=f(x , y) is said to be continuous at
the point (x₀ , y₀), if for all points (x , y) near (x₀ , y₀)
the value of f(x , y) differs , but little , from value
f(x₀ , y₀). In the other words if f has the domain R
and Q = (x₀ , y₀) is a point of R, then f is continuous
at Q if for every ε>0 there exists a δ>0 such that
|f(P) – f(Q)| = |f(x , y) – f(x₀ , y₀)| < ε

11. • For all P = (x , y) in R such for which
d((x , y),(x₀ , y₀)) = √[(x-x₀)² + (y-y₀)²]< δ
• Let f be a function of two variables x and y & (x₀ ,
y₀) be a point of R² , then f is continuous at (x₀ , y₀)
if,
limit f(x , y) = f (x₀ , y₀)
(x , y)→ (x₀ , y₀)

12. EXAMPLES
• 1.] f(x , y)= 2xy/x²+y² , if (x , y)≠(0 , 0)
= 0 , if (x , y)=(0 , 0)
→ lim
(x , y) →(0 , 0) [2x(mx)/x²+y²]
y=mx
=2m/(1+m²)
Which only depends on m.
So the limit D.N.E.
So the function can’t be continuous at (0 , 0).

13. • 2.]
f(x , y) = x²-y²/x²+y² , if (x , y) ≠ (0 , 0)
= 0 , if (x , y) = (0 , 0).
→ lim
(x , y) →(0 , 0) [x²-m²x²/x²+m²x²]
y=mx
=(1-m²)/(1+m²)
Which only depends on m.
So the limit D.N.E.
So the function can’t be continuous at (0 , 0).

14. THANK YOU
• PRESENTED BY :-
• BHAGYESH PATEL(22)
• UTKARSH GANDHI(23)
• KUNAL PATIL(24)
• GUIDED BY :-
• SWAGAT SIR