z-axis SUBMITTED BY: Ms. Harjeet Kaur Associate Professor Department of Mathematics PGGCG 11, Chandigarh y-axis x-axis

Transcription

1 z-ais - - SUBMITTED BY: - -ais ais Ms. Harjeet Kaur Associate Proessor Department o Mathematics PGGCG Chandigarh

2 CONTENTS: Function o two variables: Deinition Domain Geometrical illustration o some unctions Limit o a unction o two variables: Deinition Application o deinition o it in analzing a unction Properties o Limits o Functions o two variables Iterated Limits or Repeated Limits Two-Path Test or Non-Eistence o Limit Test o Eistence o Limit using convergence o sequences Continuous unction o two variables: Deinition Analzing continuit o a unction

3 FUNCTIONS OF TWO VARIABLES: A real valued unction o two real variables and is a rule which associates a unique real number to ever possible ordered pair o real numbers. We usuall write z = and are called independent variables and z is called dependent variable. Geometricall z = is a surace in space. Eample: Let then is a real unction o two real variables and deined on entire XY-plane ecept the points on the line + =.

4 So the given igure clearl indicates that the unction is not deined or points satising equation + =. This is m graph 4 z-ais ais ais

5 Let Here is a unction o two real variables. For a DOMAIN o : Thus the domain o this unction consists o those points o the XY-plane which are interior to the circle o radius 5 units having origin as the centre & eterior to the circle o radius units having origin as the centre. 5 log 5 5 R ILLUSTRATION:

6 LIMIT OF A FUNCTION OF TWO VARIABLES: We sa a unction approaches the it L as approaches and write: L I or ever number > there eists a corresponding > such that or all in the domain o L < whenever < < The deinition o it sa that the distance between and L becomes arbitraril small whenever the distance rom to is made suicientl small.

7 Find i it eists: We observe that along the line = the unction alwas has value when and along the line = the unction alwas has value when. So i the it eists it must be. Let > be an arbitrar number selecting So b deinition = ais -ais z-ais APPLYING THE DEFINITION OF LIMIT : whenever.. e i

8 Properties o Limits o Functions o Two Variables: The ollowing rules hold i L M and k are real numbers and Sum Rule: Dierence Rule: Product Rule: Constant Multiple Rule: an number k L M g M L g M L g M L g kl k and

9 Quotient Rule: ;M. Power Rule: I r and s are integers with no common actors and s then provided is a real number. I s is even we assume that L > s r L s r s r L M L g

10 ITERATED LIMITS or REPEATED LIMITS: The its denoted b and denoted b are called ITERATED LIMITS. For the sake o distinction is called SIMULTANEOUS LIMIT. The two iterated its i the eist need not necessaril be equal. I the two iterated its eist and are equal there is no guarantee o eistence o.

11 CONTINUED. ILLUSTRATION: Consider ; Let along the line = k ; k є R which is not unique as it takes dierent values or dierent values o k. So the two iterated its eist but the simultaneous it does not eist. k k

12 z-ais Two-Path Test or Non-Eistence o Limit: I a unction has dierent its along two dierent paths as ; then does not eist. For a it to eist at a point the it must be the same along ever approach path. Analzing the behavior o it o unction as approaches. 4.5 Checking the it along the path k ; k R. k 4 k k k ais ais So the above it depends upon the path o approach. LIMIT DOESN T EXIST.

13 Test o Eistence o Limit using convergence o sequences: Let be a real valued unction deined on a non-empt subset o ; and be a it point i ever deleted neighbourhood o contains a point o S o then i the sequence o reals converges to or all sequences in converging to in. D D D {} R R } { m l } { n l ILLUSTRATION: } : { R D Let = i i = then does not eist. The sequence in D - converges to & converges to. The sequence in D - converges to & converges to. m m m m

14 CONTINUOUS FUNCTION OF TWO VARIABLES: A unction is continuous at the point i: is deined at eists A unction is continuous i it is continuous at ever point o its domain.

15 z-ais Analzing the continuit o the given unction at origin: ; = ; = ais ais

16 z-ais SOLUTION b the deinition o continuit: - = Let Selecting such that or > be an arbitrar small number ais ais i.e. we have;. Thus; is continuous at.

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