Asymptotes | Horizontal and Verticle

Lets know what is asymptotes with example.

"A straight line l is called an asymptote for a curve c if the distance btween l and c approaches zero as the distance moved along l (from some fixed point on l)tends to infinity."

"The curve c can approach asymptote l as o e moves along l in one direction or in the opposite direction or in both the directions."

"Suppose the equation y=f(x) of c is such that y is real and y-->a as x-->∞ or x-->-∞, then y=a is a horizontal asymptote. For the distance between the curve and the straight line y=a is y-a and this approaches zero as x-->∞ or x-->-∞."

"If the equation of c is such that y is real and y-->∞ or -∞ as x-->a from one side then the straight line x=a is a verticle asymptote. To see this observe that (i)x-a is the distance between the curve and the straight line and that this distance is supposed to approach zero (ii)y-->∞ or -∞ as x-->a so that 

Lim           =    lim   (x-a) = 0

 y-->+-∞        x-->a

Thus to locate vertical asymptotes we have to find a number a such that lim     y = ∞ or - ∞."

                                                       x-->a

"Similarly if y-->mx+c as x-->∞ or -∞ then y=mx+c is an asymptote( which is neother vertical nor horizontal."

"Thus we inquire for lim xapproaches to + ,- infinity y in studying asymptote."

"For algebraic equations we can dind horizontal and vertical asymptotes as follows:

For horizontal asymptotes we write the given equation in the form x= Ψ(y)/ θ(y) and consider those values of y for which theta y is 0."

"Similarly to find a vertical asymptote we write the given equation in the form y=f(x)/g(x) and consider those values of x for which g(x)=0."

Working rule for asymptotes parallel to the axes:  "In an equation of a curve the coefficient of the highest power of x (respectively of y) equated to zero gives asymptotes (if any) parallel to the x axis (respectively y axis)."

Lets consider an example.

Example:

Lets find the asymptote of the curve y=1/(x-2)²

Solution:

When x-->2 from either side ,y is positive and approaches ∞ . Thus x=2  is a verticle asymptote.

Again lim     y = 0     thus y=0 is a horizontal 

             x-->∞ 

asymptote .

Thus asymptote can be readily obtained by the working rule stated above.

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