definitely foreign b foreign foreign cause needed a fundamental theorem of limits fundamental funda mental theorems mental theorem theorems on limits are [Laughter] forms right challenging problems everything indeterminate forms basic limits standard limits this is a uniform process uniform strategy anything anything in the order so fundamental theorems on limits to basic fundamental theorems on limits so fundamental theorems on limits change absolutely suppose f of x equal to l limit x tends to a f of x equal to l limit x tends to a g of x is equal to m anakum where l comma m r l comma m are finite numbers limit extends to a f of x plus leather minus g of x in a split change limit can be applied to f of x so a n does not limit extends to a f of x plus r minus limit x tends to a g of x very good extends to a f of x into moly limit x tends to a g of x so product alone program mirror limit is split but limit x tends to f of x and t over to l into limit extends to a g of x and a m so l into m um third rule third row limit x tends to a x tends to a all right limit extends to a f of x by g of x equal to uh division loan of program mirror limit applies to limit x tends to a f of x by limit x tends to a g of x ah limit x tends to a f of x and a l limit extends to your g of x and t m d m zero coordinates right fundamentally m [Music] foreign h for example limits uh chapter one i limit extends to zero x square sine uh uh one by x under quantity sign on by exit london then limit mono directly up to limits finding finding limits finding limits low finding limits low monica render column and so is limit so the foreign x tends to minus 2 x square plus 3 x minus 2 approaches 2 whole square plus 3 4 plus 3 that equal to 7 so it is 7 approach although even a direct substitution problems direct substitution problems so but direct substitution limit x tends to 1 x square minus 1 by so 1 square into 1 a 1 minus 1 by 1 minus 1 0 by 0 0 by 0 is indeterminate value and define undefined indeterminate value and you find out about the indeterminate value [Applause] factorization factorization render rationalization rationalization standard 0 my so x square minus a square by x minus yellow limit x tends to a x square minus a square knee x minus a into x plus a and by foreign [Music] plus 2 into 1 plus 3 so 1 plus 2 plus 3 6 foreign x tends to one x square plus limit x tends to one to x plus limit x tends to one three at a limit speed just first fundamental theorem of f of x plus g of x nancy 0 minus 3 into 0 plus 2 so 1 by e 0 e 0 is 2 so i pin 1 by 2 so 0 approaches the function 1 by 2 approaches right next to x 3 and approaching the function approach 2 into 1 plus 1 by 3 into 1 square minus 4 into 1 plus 5 2 plus one three by three plus five eight eight minus four four three by four and approaching okay direct substitution possible end so either one approach within the e function over an approach only one square plus two by one square minus 2 so 3 by minus 1 so that is minus 3 antenna 1 plus 2 3 1 minus 2 minus 3 see 1 minus 2 minus 1 3 by minus 1 minus 3 right e functions 2 by three minus three by two two twos are four minus three threes are nine by six so minus five by six uh addition subtraction minus one by zero square plus four zero minus one minus one by four so direct substitution equals okay zero 0 0 infinity by infinity zero into negative indeterminate forms you can understand very very important sand which is any angle of course cos 90 degrees cost 30 degrees cos 60 degrees cos 1 degree 2 degree 4 degrees cos of anything will lies between minus 1 and 1 trigonometry luminates your values square into cos 2 by x so you taught a line equation x square to multiply j so minus 1 into x square e x square into cos 2 by x minus x square less than or equal to x square cos 2 by x less than or equal to x square e function is limit x tends to zero minus x square and then zero substituted independent formula together so zero substituted just a minus zero square root of zeros [Music] limit x tends to zero x square zero zero substitution zero square function zero which in the limit function is theorem limit extends to zero x square cos two by x is also zero so ceramic in limit all of another 3 whole square minus 9 by 3 cube minus 6 into 3 square plus 9 into 3 plus 1 3 square under 9 minus minus 27 minus 3 square 9 9 6 are 54 plus 9 3's are 27 plus one in the zero rod like a 27 plus 27 54 minus 54 cancel out in the one mega thing so zero by one mega in the anti zero okay indeterminate form of the capital okay function zero minus 1 by 1 minus 1 1 square root of 1 minus 1 minus 1 foreign one x minus 1 by x square minus x minus then limit 1 by x cube minus 3 x square plus x character 1 substituting x minus one x commonly taken 2 minus 3 under 1 minus 1 by minus 1 1 plus 1 and they get the minus 1 and e minus plus sigma 1 plus 1 2 so it wattal function 2 in approaching foreign minus 81 by 2 into 3 square minus 5 into 3 minus 3 81 minus 81 numerator 0 i put in the three power 481 denominator equal to zero at the independent formation three square nine nine twos are eighteen minus five threes are fifteen minus three i penny fifteen three eighteen eighteen minus foreign is square minus 6x plus 1x minus 3 uh i'll put the x square minus 9 into x square plus 9 although a square minus b square x square minus 3 square me x minus 3 into x plus 3 into x square plus 9 so v lineup of factorization in the numerators factors 2x plus 1 into x minus 3 so x minus 3 x minus 3 can slip in the iguana megaline is limits right indicator limit excellence to 3 yeah limit x tends to three limit x tends to three so maybe not like promoting substitution indeterminate formulas three x three and approaching 3 plus 3 6 into 3 square 9 9 plus 9 18 by 3 substitutions three twos are six plus one seven cancellations 24 plus 15 by 3 square 9 minus 9 so 9 plus 15 24 minus 24 0 in so limit x tends to 3 x square minus three x minus five x plus fifteen that's pretty much along the eight x x square minus nine at x minus three into x plus three right limit x tends to three x minus three into x minus five since x minus three into x plus three cello x minus x minus cancel so indeterminate formula substituted to 3 minus 5 by 3 plus 3 so minus 2 by 6 them equal one substituted minus minus root law twenty five minus one one square into one a so minus root twenty four e f of one and f of x is limit x tends to 1 f of x minus root 25 minus x square minus question low minus i put f of 1 f of one whatever minus root twenty four so minus root twenty four jagger already minus the question minus by x minus minus root 25 minus x square minus into minus plus root 24 tends to one root 24 minus root 25 minus x square by x minus 1 right x 1 and minus root 25 minus 190 24 by x one negative one minus one i penny zero in the zero index zero by zero formula within s 24 plus root 25 minus x squared ah limit x tends to 1 ah a minus b into a plus b member mode on the a square minus b square a square under root 24 square under 24 mu minus b square root of 25 minus x square k whole square root square root of both 25 minus x square minus 1 bracket better much probably minus by x minus 1 into root 24 plus root 25 minus x square repair in kypolis servicing as a root only 24 minus 25 plus x square and then minus the openness by x minus 1 into root 24 plus root 25 minus x square 24 minus 25 minus one of the minus so x square minus 1 out of the numerator x square mundra is minus 1 by x minus 1 into 24 plus root 24 plus 25 minus x x minus 1 into root 24 plus root 25 minus x square so x minus 1 x minus 1 cancel indeterminate form there is limit x tends to 1 root 24 minus root 25 minus x square by x minus so laptop rule of nomination differentiate yes numerator and differentiate your sum denominator and differentiate just mean optimal constant so minus minus root x equal differentiation 1 by 2 root x 1 by 2 root x root x equal differentiation 1 by 2 root x into 25 minus x square internal differentiation minus 2x system differentiations i mean differentiation internal differentiation json so numerator total differentiation in the denominator differentiation foreign keep watching happy learning bye-bye hindi