Putting y = mx+c . . . .(1) in the equation involving x and y only and with total degree of each term at the most 3, we get a polynomial of degree 3 in x only with two parameters m and c which we find for (1) to be an asymptote of the given curve. Dividing out by x³ we get a term independent of x, while all the other terms have powers of x in the denominator and they approach 0 as x tends to infinity. Thus essentially we are left with the coefficient having equated to 0, giving us one relation involving m and possibly c also.
Again now removing the x³ term, we divide by x² and again by an argument similar to above, we get the coefficient of x² having been equated to 0. The above two relations can be used to determine m and c for the equation (1) to be an asymptote i.e. as a line at a finite distance from the origin which approaches the given curve as we go farther and farther from the origin.
This method finds all the asymptotes except those parallel to the y-axis.
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Answer:
Putting y = mx+c . . . .(1) in the equation involving x and y only and with total degree of each term at the most 3, we get a polynomial of degree 3 in x only with two parameters m and c which we find for (1) to be an asymptote of the given curve. Dividing out by x³ we get a term independent of x, while all the other terms have powers of x in the denominator and they approach 0 as x tends to infinity. Thus essentially we are left with the coefficient having equated to 0, giving us one relation involving m and possibly c also.
Again now removing the x³ term, we divide by x² and again by an argument similar to above, we get the coefficient of x² having been equated to 0. The above two relations can be used to determine m and c for the equation (1) to be an asymptote i.e. as a line at a finite distance from the origin which approaches the given curve as we go farther and farther from the origin.
This method finds all the asymptotes except those parallel to the y-axis.