 Hello and welcome to the session. In this session we will discuss about asymptotes. Now let us consider the function y is equal to 1 upon x. Let us see its graph. We see that there are two curves in its graph. One lies in first quadrant and the other in third quadrant. Now we see that the branches of these curves are becoming closer and closer to y-axis but they never intersect or cross y-axis. It means that y values of the function approach to plus infinity and minus infinity as x approaches to 0. So we say that y-axis is the asymptote of the given function. So we define asymptotes as an asymptote is a straight line which acts as a boundary for the graph of the function. When our function has an asymptote the output values that is y values of the function get closer and closer to the asymptote as the input values that is the x values approach to either a specific value or positive or negative infinity the functions most likely to have asymptotes are the rational functions. We will discuss three types of asymptotes and these are vertical asymptote horizontal asymptote and slant asymptote. First of all we shall discuss vertical asymptote. Now in the graph of the function y is equal to 1 by x, y-axis is a straight vertical line and it is the vertical asymptote of this function and its equation is given by x is equal to 0. See as x approaches to 0 from right the value of f of x or y increases without bounds towards plus infinity and as x approaches to 0 from left the value of f of x decreases without bound towards minus infinity. So we define the line x is equal to c is the vertical asymptote for a function f of x if f of x tends to plus infinity or f of x tends to minus infinity as x tends to c from either the left or from the right. Now we see how to find vertical asymptote as a brightly. Let us consider the rational function f of x is equal to p of x upon q of x. The vertical asymptotes are found by equating the denominator of the function equal to 0 and solving it for x that is we put q of x is equal to 0 and solve it for x the obtained values of x will give the vertical asymptote for example in the function f of x is equal to 1 by x we know that it is a rational function where q of x is equal to x so we put q of x is equal to 0 which implies that x is equal to 0 so here value of x is 0 so vertical line is drawn at x is equal to 0 which is the vertical asymptote of the function. Now we are going to discuss horizontal asymptote again see the graph of y is equal to 1 by x c in first quadrant as the value of x increases or approaches to infinity the value of y becomes smaller and smaller and approaches to 0 similarly in third quadrant as x approaches to minus infinity by approaches to 0 so these branches of the curve become closer and closer to x axis but do not intersect or cross x axis thus here x axis is the horizontal asymptote whose equation is y is equal to 0 thus we define the line y is equal to d is the horizontal asymptote for our function f of x if f of x tends to d as x tends to plus infinity or x tends to minus infinity now let us see method of determining horizontal asymptote as a brightly in our rational function f of x is equal to p of x upon q of x we determine the horizontal asymptotes by looking at the degree of both numerator and the denominator of the given function if degree of numerator is less than degree of denominator then the horizontal asymptote is always the line y is equal to 0 for example in the function y is equal to 1 by x the degree of numerator is 0 and degree of denominator is 1 here we see that the degree of numerator is less than the degree of denominator so here horizontal asymptote will be equal to the line y is equal to 0 if degree of numerator is equal to degree of denominator then the horizontal asymptote is obtained by dividing the leading coefficient of the numerator by the leading coefficient of the denominator for example in function y is equal to 5x square minus 3x plus 1 whole upon 2x square plus 5 the degree of both numerator and denominator is 2 leading coefficient is the coefficient of the highest power of x so in the numerator the leading coefficient is 5 and it is 2 in denominator so horizontal asymptote will be given by y is equal to 5 upon 2 so on the graph we will draw horizontal line at y is equal to 5 upon 2 also we should note that if degree of numerator is greater than degree of denominator then we get a slant line as the asymptote and it is called slant asymptote now let us see the method how to determine equation of slant asymptote to find the equation of slant asymptote we divide numerator by denominator of the given function using long division now let us consider the function y is equal to x square minus 3 whole upon x plus 1 here degree of numerator is greater than degree of denominator we see that here degree of numerator is 2 and degree of denominator is 1 and 2 is greater than 1 so we say that degree of numerator is greater than degree of denominator so using long division we divide numerator by denominator and we get now here x into x is x square plus 1 into x is plus x on solving we get minus of x minus 3 now again we multiply x by minus 1 and we get minus x plus 1 into minus 1 is minus 1 and on solving further we get minus 3 plus 1 that is minus 2 so here the quotient is x minus 1 which is the required asymptote and therefore we say that the equation of slant asymptote is given by y is equal to x minus 1 here we have seen we can write x square minus 3 as x minus 1 the whole into x plus 1 the whole minus 2 because we can write dividend is equal to quotient into divisor plus remainder now dividing both sides by x plus 1 the whole we get x square minus 3 the whole upon x plus 1 which is equal to x minus 1 the whole into x plus 1 the whole upon x plus 1 minus 2 upon x plus 1 which implies that x square minus 3 the whole upon x plus 1 is equal to x minus 1 the whole minus 2 upon x plus 1 now when x approaches to infinity then 2 upon x plus 1 approaches to 0 so we are left with x square minus 3 the whole upon x plus 1 is equal to x minus 1 now this is the given function and it can be written as y is equal to x minus 1 so now we will draw the line y is equal to x minus 1 as the asymptote of the given function now this is the graph of the function y is equal to x square minus 3 whole upon x plus 1 this glance line shows the asymptote of the function which is given by the equation y is equal to x minus 1 now see the branches of the curve are becoming closer and closer to the straight line y is equal to x minus 1 thus in this session we have discussed about asymptote and its types this completes our session hope you enjoyed this session