Limits at Infinity; Horizontal Asymptotes Definition: Let f be a function defined on some interval (a, ∞). Then Lxfx=∞→)(lim means that the value of f (x) can be made arbitrarily close to L when x becomes sufficiently large. Definition: Let f be a function defined on some interval (−∞, a). Then Lxfx=∞−→)(lim means that the value of f (x) can be made arbitrarily close to L when x becomes sufficiently large negative. Ex. xxxx1lim01lim∞−→∞→== Ex. Neither xxsinlim∞→ nor xxcoslim∞→ exists. Definition: For a real number* L, the line y = L is a horizontal asymptote of the curve y = f (x) if either Lxfx=∞→)(lim or Lxfx=∞−→)(lim * That is, L is a finite number; recall that ∞ or −∞ are not real numbers. Note: Therefore, the graph of a function can have at most 2 horizontal asymptotes. Ex. The line y = 0 is the horizontal asymptote of xxf1)( =Theorem: If r > 0 is a rational number, then ∞=∞→rxxlim Ex. ∞==∞→∞→10/110limlim xxxx Theorem: If r > 0 is a rational number, then 01lim =∞→rxx If r > 0 is a rational number such that xr is defined for all x, then 01lim =∞−→rxx Ex. 01lim1lim2/1==∞→∞→xxxx Ex. 0)0(51lim55lim3/13===∞−→∞−→xxxx Limits at Infinity of Rational Functions: According to the above theorem, if n is a positive integer, then nxnxxx1lim01lim∞−→∞→== This fact can be used to find the limits at infinity for any rational function.Ex. 0101000100lim32=−++∞→xxxx The limit as x → −∞ is the same. The graph of the function in the limit above has a horizontal asymptote y = 0. Ex. 352103785lim3424−=−+−+−∞→xxxxxx Therefore, the graph of the function in the limit above has a horizontal asymptote y = −5/3. (What is the limits as x → −∞?) Ex. ∞=−−+−∞→12544lim223xxxxx and −∞=−−+−∞−→12544lim223xxxxxIn general, the limits of rational functions at infinity can be summarized in the following set of rules: Let )()()(xgxfxr = be a rational function where f(x) is a polynomial of degree m and g(x) is a polynomial of degree n. (i.) If m < n, then 0)(lim)(lim==∞−→∞→xrxrxx (ii.) If m = n, then baxrxrxx==∞−→∞→)(lim)(lim where a and b are, respectively, the leading coefficients of f (x) and g(x). (iii.) If m > n, then )(lim xrx ∞→ and )(lim xrx ∞−→ can be either ∞ or − ∞. (The two limits don’t have to be equal, but each will be an infinite limit of some type.) Ex. 11223lim22−=−−+∞→xxxx Ex. −∞=+−∞→623lim25xxx and ∞=+−∞−→623lim25xxx Ex. −∞=−∞⋅∞=−=−∞→∞→)()1(lim)(lim343xxxxxxEx. ===∞−→∞→xxxx1sinlim0)0sin(1sinlim, and similarly ==∞−→∞→xxxx1coslim11coslim Ex. Evaluate xxxsin1lim∞→ Since the range of sin x is between −1 and 1, the inequalities hold for all values of x: xxxx1sin11≤≤−. Apply the Squeeze Theorem and we see that 01limsin1lim1lim0 =≤≤−=∞→∞→∞→xxxxxxx. Hence, 0sin1lim =∞→xxx. Its graph: Note the removable discontinuity at x = 0.Ex. Evaluate ∞→xxx1sinlim Ans. 1 Ex. )(lim22xxxxx−−+∞→= 1 Ex. Evaluate 192lim2++∞→xxx and 192lim2++∞−→xxx Ex. Evaluate ()xxxx2lim2++∞−→ Ans. −1 Ex. (Math 141 exam 2, spring 2005) Evaluate ()xxxx−+∞→3lim2 Ans. 3/2Summary of Curve Sketching Steps to take / things to find when sketching the curve y = f (x): 1. Domain of the function 2. Intercepts 3. Symmetry 4. Asymptotes 5. Interval of Increase / Decrease 6. Local Maximums and Minimums 7. Concavity and Inflection Points 8. Sketch the Curve Ex. Graph y = x3 − 3x + 3 Ex. Graph 1122−+=xxyOblique (Slant) Asymptotes An oblique or slant asymptote is an asymptote that’s neither vertical nor horizontal. For rational functions, it exists on the graph whenever the degree of the numerator is exactly one higher than the degree of the denominator. Why? Suppose )()()(xgxfxr = is a rational function such that f has degree n + 1 and g has degree n. Then r(x) can be rewritten, by dividing f by g, as )()()()()()()()(xgxRxQxgxRxgxQxr +=+= such that the quotient Q(x) is a polynomial of degree 1. Its curve is a line and it becomes the oblique asymptote on the graph of r(x). The reason is that the remainder R(x) has degree less than n, so the limit of the quotient R(x)/g(x) goes to 0 as x → ±∞. Hence, r(x) → Q(x) as x → ±∞, and therefore y = Q(x) is the oblique asymptote. Ex. xxxxf12)(2++= Dividing x2 + 2x + 1 by x, we get xxxf1)2()( ++=. Therefore, the graph has an oblique asymptote y = x + 2. It also has a vertical asymptote x = 0.The graph of xxxxf12)(2++=:Optimization Goal: To apply the techniques of finding maximum/minimum to find ways to maximize benefit or minimize cost/material, etc. Ex. What is the largest rectangular area that can be enclosed by 20 yards of fence? Ans. 25 square yards Ex. 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Ans. 4000 cm3 Ex. Find 2 positive numbers whose product is 100 and whose sum is a minimum. Ans. 10 and 10 Ex. A farmer has 240 yards of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? Ans. 60 × 120 Ex. A cylindrical can is to hold 1000π inch3 of water. Different material is used for top/bottom lids and the side − the lids cost 3 cents/inch2 and the side costs 6 cents/inch2. Find the minimum cost of the can. Ans. 1800π cents ≈ $55 Ex. What is the largest volume that a lid-less rectangular box, made out of 5ft × 8ft sheet-metal, can have? Ans. 18 ft 3Ex. Find the point on the line y = 2x − 4 that is closest to the point (1, 3). Ans. the closest point is (3, 2) Ex. What is the area of the largest rectangle that can be inscribed in a circle of radius 1? Ans. area =