Instruction

1

Find the vertical asymptotes. Given a function y=f(x). Find the definition area and select all the point a where the function is not defined. Calculate the limits lim(f(x)) when x tends to a, (a+0) or (a−0). If at least one limit is +∞ (or -∞), vertical asymptote of the graph of the function f(x) be the line x=a. Calculating two one-way limit, you determine how a function behaves when approaching the asymptote from different directions.

2

Explore a few examples. Let the function y=1/(x2−1). Calculate the limits lim(1/(x2−1)) when x tends to (1±0), (-1±0). The function has vertical asymptotes x=1 and x=-1 since the limits equal to +∞. Let given the function y=cos(1/x). This function has no vertical asymptotes x=0, since the region of change of a function the cosine of the segment [-1; +1] and its limit will never be equal to ±∞ for all values of x.

3

Now find the oblique asymptote. To do this, compute the limits k=lim(f(x)/x) and b=lim(f(x)−k×x) at x tending to +∞ (or -∞). If they exist, then the oblique asymptote of the graph of the function f(x) is set by the equation of the line y=k×x+b. If k=0, the line y=b is called horizontal asymptote.

4

Consider, for better understanding the following example. Let given the function y=2×x−(1/x). Calculate the limit lim(2×x−(1/x)) when x tends to 0. This limit is equal to ∞. That is a vertical asymptote of the function y=2×x−(1/x) be the line x=0. Find the coefficients of the equation of a slant asymptote. To do this, calculate the limit k=lim((2×x−(1/x))/x)=lim(2−(1/x2)) for x tending to +∞, so k=2. Now, calculate the limit b=lim(2×x−(1/x)−k×x)= lim(2×x−(1/x)-2×x)=lim(-1/x) for x tending to +∞, that is b=0. Thus, the oblique asymptote of this function is given by the equation y=2×x.

5

Please note that the asymptote may intersect the curve. For example, for the function y=x+e^(-x/3)×sin(x) limit lim(x+e^(-x/3)×sin(x))=1 for x tending to ∞, but lim(x+e^(-x/3)×sin(x)−x)=0 for x tending to ∞. That is, the asymptote is the line y=x. It intersects the graph of the function at several points, for example, at the point x=0.

Note

The sign ^ denotes exponentiation.