<p><span style="font-size: 1rem;">Lets know what is asymptotes with example.</span><br></p>
<p><span style="font-size: 1rem;">"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."</span><br></p>
<p>"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."</p>
<p><br></p>
<p>"Suppose the equation y=f(x) of c is such that y is real and y--&gt;a as x--&gt;∞ or x--&gt;-∞, 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--&gt;∞ or x--&gt;-∞."</p>
<p><br></p>
<p>"If the equation of c is such that y is real and y--&gt;∞ or -∞ as x--&gt;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--&gt;∞ or -∞ as x--&gt;a so that&nbsp;</p>
<p>Lim&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=&nbsp; &nbsp; lim&nbsp; &nbsp;(x-a) = 0</p>
<p>&nbsp;y--&gt;+-∞&nbsp; &nbsp; &nbsp; &nbsp; x--&gt;a</p>
<p>Thus to locate vertical asymptotes we have to find a number a such that lim&nbsp; &nbsp; &nbsp;y = ∞ or - ∞."</p>
<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;x--&gt;a</p>
<p><br></p>
<p>"Similarly if y--&gt;mx+c as x--&gt;∞ or -∞ then y=mx+c is an asymptote( which is neother vertical nor horizontal."</p>
<p><br></p>
<p>"Thus we inquire for lim xapproaches to + ,- infinity y in studying asymptote."</p>
<p><br></p>
<p>"For algebraic equations we can dind horizontal and vertical asymptotes as follows:</p>
<p><br></p>
<p>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."</p>
<p>"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."</p>
<p><br></p>
<p>Working rule for asymptotes parallel to the axes:&nbsp; "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)."</p>
<p>Lets consider an example.</p>
<p>Example:</p>
<p><br></p>
<p>Lets find the asymptote of the curve y=1/(x-2)²</p>
<p><br></p>
<p>Solution:</p>
<p><br></p>
<p>When x--&gt;2 from either side ,y is positive and approaches ∞ . Thus x=2&nbsp; is a verticle asymptote.</p>
<p><br></p>
<p>Again lim&nbsp; &nbsp; &nbsp;y = 0&nbsp; &nbsp; &nbsp;thus y=0 is a horizontal&nbsp;</p>
<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;x--&gt;∞&nbsp;</p>
<p>asymptote .</p>
<p><br></p>
<p>Thus asymptote can be readily obtained by the working rule stated above.</p>
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