The full research of function and creation of its schedule assume the whole range of actions, including finding of asymptotes which are vertical, inclined and horizontal.
Instruction
1. Asymptotes of function are applied to simplification of creation of its schedule and also a research of properties of her behavior. The asymptote is a straight line which the infinite branch of the curve set by function approaches. Distinguish vertical, inclined and horizontal asymptotes.
2. Vertical asymptotes of function are parallel to ordinate axis, these are straight lines of a look x = x0 where x0 is a boundary point of a range of definition. The point in which unilateral limits of function are infinite is called boundary. To find asymptotes of this sort, it is necessary to investigate her behavior, having calculated limits.
3. Find a vertical asymptote of the f (x) function = x² / (4 • x² - 1). For a start define its range of definition. It can be only value at which the denominator addresses in zero, i.e. solve the equation 4 • x² – 1 = 0 → x =±1/2.
4. Calculate unilateral limits: lim _ (x →-1/2) x² / (4 • x² - 1) = lim x² / ((2 • x - 1) • (2 · x + 1)) = + ∞. lim _ (h1/2) x² / (4 • x² - 1) = - ∞.
5. Thus, you found out that both unilateral limits are infinite. Therefore, straight lines x =1/2 and x =-1/2 are vertical asymptotes.
6. Inclined asymptotes are straight lines of a type of k • x +b, in which k = lim f/x and b = lim (f – k·x) at x →∞. Such asymptote will become horizontal at k=0 and b ≠∞.
7. Learn whether function from the previous example has inclined or horizontal asymptotes. For this purpose define coefficients of the equation of a direct asymptote through the following limits: k = lim (x² / (4 • x² - 1)) / x = 0; b = lim (x² / (4 • x² - 1) – k·x) = lim x² / (4 • x² - 1) = 1/4.
8. So, this function has also an inclined asymptote and as the condition of zero coefficient of k and b not equal to infinity is satisfied, it is horizontal. Answer: function x² / (4 • x² - 1) has two vertical x = 1/2; x =-1/2 and one horizontal at = 1/4 asymptotes.