Definition of an Asymptote: a Line
-Vertical Assymptotes (V.A.): A vertical line where the function is undefined
Example: Graph on calculator 1/(x-2), the V.A. x=2
- Horizontal Assymptotes: A horizontal line that the func. approaches, the limit as x→+/-∞
*The function. can go thru the H.A., the point is to find the limiting value of the function as x gets very large.
Example: f(x)=1/x (graph on calc. to see the function) lim x→∞=0, lim x→-∞=0. So there is a H.A. at y = 0.
lim x→0 = DNE (-/+∞). There is a V.A. at x = 0.
H.A.: the line the function approaches as x goes to +/-∞ ( if you follow the function with your finger as it approaches the horizontal line, that is your H.A.)
-If the function is a Rational function (a polynomial over a polynomial), then if the denominator can’t factor there is an infinite discontinuity.
Example. lim x→-2 f(x)=(x+1)/(x-3)(x+2) →the denominator approaches 0. Since we can’t cancel any factors there are two infinite discontinuities.
x→-2‾ (x+1)/(x-3)(x+2) →-∞
x→-2+ (x+1)/(x-3)(x+2) → +∞
So, from these choose a value close to the right or left of the value that x is approaching, as the case may be, and plug it in to find the sign of each factor, then use that to determine the signs of the whole function to determine if it is approaching the – or + ∞, which is the limit.
-Finding H.A.
-find the lim as x→∞
Example: f(x )= 2-(1/x)
lim x→∞f(x)= 2- (1/x approaches 0 as the denominator gets closer to ∞)
so, lim =2 (Which is the H.A)
*****Rule******
lim x→-/+∞ 1/x^n = 0
Example: lim x→∞ (5x-7)/(4x+3)= +∞/+∞ ≠1 → infinity divided by infinity is an indeterminate form
* divide each term by x (the largest exponent of x in the denominator), so the problem becomes limx→∞[(5-7/x)/ (4+3/x)]
- so the n/x fraction approaches 0
-so 5/4 is the H.A. ( where x→∞)
So, when x approaches ∞ with 2 polymonials divided by each other, as long as the power of the highest term of each is the same, the H.A. is the coefficients of the highest terms
Example: lim x→∞ (3x^2 +7x +1)/(2x^2 + 3) → multiply by 1/x^2 → (3+ 7/x +1/x^2)/(2+3/x^2)
so, lim x→ ∞ = 3/2
***************Rule**************
the mathematical statement of the above can be found in theorem 5.2 on page 113
Examples 5.12 in the book is good example on page 117
