 Hello and welcome to the session. In this session first we will discuss limits. If we have that as x tends to a fx that is the given function tends to l then this l is called the limit of the function fx and this is symbolically written as limit as x tends to a fx is equal to l. There are two ways in which x could approach a number a either from left or from right that is all the values of x near a could be less than a or could be greater than a. So this leads to two limits the right hand limit and the left hand limit limit x tends to a plus fx this is the expected value of f at x equal to a given the values of f near x to the right of a and this value is called the right hand limit of fx at a then limit x tends to a minus fx is the expected value of f at x equal to a given the values of f near x to the left of a this value is called the left hand limit of fx at a then the limit of a function at a point is the common value of the left and right hand limit if they coincide and is denoted by limit x tends to a fx this is the limit of the function fx at x equal to a this happens when the left hand limit and the right hand limit of the given function at x equal to a coincides consider the function fx given by 2x plus 3 then x is less than equal to 0 and 3 into x plus 1 when x is greater than 0 we need to find limit x tends to 0 fx first let's find the left hand limit of fx at x equal to 0 that is limit x tends to 0 minus fx now in this case we will consider fx as 2x plus 3 because we have fx is equal to 2x plus 3 when x is less than equal to 0 so this becomes limit x tends to 0 minus 2x plus 3 then we substitute x equal to 0 minus h now as x approaches 0 h also approaches 0 so we get limit h approaches 0 2 into 0 minus h plus 3 that is this is equal to limit h tends to 0 minus 2h plus 3 now when we put h equal to 0 in this we get this to be equal to 3 so we have limit x tends to 0 minus fx is equal to 3 that is the left hand limit comes out to be equal to 3 now let's consider the right hand limit that is limit x tends to 0 plus fx now in this case we will take fx as 3 into x plus 1 as x is greater than 0 in this case so we get this is equal to limit x tends to 0 plus 3 into x plus 1 now here we will substitute x as 0 plus h as x tends to 0 h also tends to 0 so we get limit h tends to 0 3 into 0 plus h plus 1 now we simply substitute h equal to 0 in this and we get this equal to 3 that is we get limit x tends to 0 plus fx is equal to 3 this is the right hand limit so we get the left hand limit is equal to the right hand limit that is equal to 3 so we get limit x tends to 0 fx is equal to 3 next we discuss algebra of limits we consider two functions f and g such that limit x tends to a fx and limit x tends to a gx exist then we have limit x tends to a fx plus gx is equal to limit x tends to a fx plus limit x tends to a gx then next is limit x tends to a fx minus gx is equal to limit x tends to a fx minus limit x tends to a gx then limit x tends to a fx into gx is equal to limit x tends to a fx into limit x tends to a gx then limit x tends to a fx upon gx is equal to limit x tends to a fx upon limit x tends to a gx and also the particular case of this third property if we have that function gx is some constant lambda that is limit x tends to a lambda fx this would be equal to lambda into limit x tends to a fx if we consider fx to be a polynomial function fx is a polynomial function if we have fx is equal to a naught plus a 1x plus a 2x square plus and so on up to an xn where ai is a real number such that an is not equal to 0 for some natural number n then limit x tends to a fx which is a polynomial function is equal to f of a and if we have the function fx to be a rational function that is fx is of the form gx upon hx where g and hx are polynomials such that hx is not equal to 0 then in that case limit x tends to a fx would be equal to g of a upon h of a if in case we get h a is equal to 0 then we get two cases when ga is not equal to 0 and when ga is equal to 0 then we have ga is not equal to 0 and h a is equal to 0 then limit does not exist and if we have ga equal to 0 and h a equal to 0 then this is of the form 0 upon 0 then in this case we try to rewrite the function that is we factorize the numerator the denominator and we cancel the factors which are causing the limit to be of the form 0 upon 0 let's try and find the limit x tends to 3 x square minus 9 upon x minus 3 now when we put x equal to 3 in the numerator and in the denominator we get the form 0 upon 0 so what we do is we factorize the numerator and the denominator so we get x plus 3 into x minus 3 upon x minus 3 we cancel x minus 3 and x minus 3 and we put x equal to a in the term which is left that is limit x tends to 3 x plus 3 when we put x equal to 3 in this we get this equal to 6 so our answer is 6 evaluation of an important limit which will be used in the sequel is given as for any positive integer n limit x tends to a x to the power n minus a to the power n upon x minus a is equal to n into a to the power n minus 1 this expression would be true even if n is any rational number and a is positive let's evaluate the limit x tends to a x to the power 12 minus a to the power 12 upon x minus a this is of the form limit x tends to a x to the power n minus a to the power n upon x minus a so this would be equal to n into n is 12 so 12 into a to the power n minus 1 that is 12 minus 1 which is equal to 12 into a to the power 11 so this is the answer now we consider limits of trigonometric functions we have a very important theorem which is used in calculating limits of some trigonometric functions which says that let f and g be two real valued functions with the same domain such that f x is less than equal to g x for all x in the domain of definition then for some a if limit x tends to a f x and limit x tends to a g x exist then limit x tends to a f x is less than equal to limit x tends to a g x another important theorem that we have is sandwich theorem according to which we have let f g and h be real functions such that f x is less than equal to g x is less than equal to h x for all x in common domain of definition for some real number a if limit x tends to a f x is equal to l is equal to limit x tends to a h x then we have limit x tends to a g x is equal to l another important limit is limit x tends to 0 psi x upon x is equal to 1 and limit x tends to 0 1 minus cos x upon x is equal to 0 let's evaluate limit x tends to 0 1 minus cos x upon x square this could be written as limit x tends to 0 now 1 minus cos x can be written as 2 sine square x upon 2 upon x square now x square can be written as x upon 2 the whole square multiplied by 4 that is this is equal to 1 upon 2 limit x tends to 0 sine x upon 2 upon x upon 2 the whole square as x tends to 0 x upon 2 also tends to 0 so this is equal to 1 upon 2 limit x upon 2 tends to 0 sine x upon 2 upon x upon 2 the whole square we know that limit x tends to 0 sine x upon x is equal to 1 so this is equal to 1 upon 2 into 1 square which is equal to 1 upon 2 so this is the answer this completes the session hope you have understood the concept of limits