MAT137 Calculus! Lecture 9 Today we will study: Limits at infinity. L Hôpital s Rule. Mean Value Theorem. (11.5,11.6, 4.1) PS3 is due this Friday June 16. Next class: Applications of the Mean Value Theorem. Extrema. Optimization Problems. (4.2-4.5)
Limits at infinity Recall the meaning of lim x c f(x) = L ε > 0, δ > 0 such that if 0 < x c < δ then f(x) L < ε. We are interested now in the behaviour of f(x) for values of x arbitrarily large (say by positive values). For instance, look at the functions f(x) = 1 x and g(x) = sin(x) x (see graphs on the board). We can see that the values of f(x) and g(x) are approaching to 0 as x.
Limits at infinity Definition Definition of limit at Let f be a function defined at least on some interval (a, ). We say that f(x) = L if lim x ε > 0, P > 0 such that if x > P then f(x) L < ε.
Limits at infinity Example 1 Let us study the limit of f(x) = 1 x when x. (see the board) 1 We can prove (exercise) that lim x x = 0 for all positive integer n. n
Limits at infinity Definition We can define the limit of f(x) as x in a similar way. Definition of limit at Let f be a function defined at least on some interval (,a). We say that lim f(x) = L if x ε > 0, N < 0 such that if x < N then f(x) L < ε.
Limits at infinity Example 2 Let us study the limit of f(x) = 1 x when x. (see the board) 1 We can prove (exercise) that lim x x = 0 for all positive integer n. n
Limits at infinity Remark The limit laws that we studied before can be adapted to the definition of (finite) limits at infinity. It is left as an exercise to formulate such limit laws.
Infinite limits at infinity Definition In some cases the function we are studying approaches to ± as x ±. Definition of infinite limit at Let f be a function defined at least on some interval (a, ). We say that f(x) = if lim x M > 0, P > 0 such that if x > P then f(x) > M. Exercise: define lim x f(x) =
Infinite limits at infinity Definition In some cases the function we are studying approaches to as x ±. It is left as an exercise to define both lim f(x) = and lim f(x) = x x
Limits at infinity Example 3 2x Find lim 3 5x+3 x 5x 4 8
Limits at infinity Example 4 2x Find lim 4 5x+3 x 5x 4 8
Limits at infinity Example 5 2x Find lim 9 5x+3 x 5x 4 8
Limits at infinity Example 6 ( Find lim x ) x 2 +25 x
Limits at infinity Example 7 Find lim 2x 2 +1 x 3x 5. Solution: homework. The answer is not 2 3.
L H^opital s rule Example 8 The indeterminate 0/0!: when computing limits of functions like f(x) g(x), as x c (or x c + or x c ), we often find out that f(x) 0 and g(x) 0 and so we are in the presence of an indeterminate 0/0. Some times such limits exist, some times they do not exist (that is why we call it indetermination). For instance, we have the following limits: sin(x) lim x 0 x, cos(x) lim x π π 2x, 2 lim x, x 0 x 2 x sin( lim x) 1 x x 0 x, lim x 0 x. Which of the previous limits exist? Only the first Two!
L H^opital s rule Indeterminate (0/0) Suppose that when x c (x c +, or x c ), we obtain that f(x) 0 and g(x) 0, but in the approach g(x) and g (x) are never 0. If f (x) f(x) g (x) γ, then g(x) γ.
L H^opital s rule Example 9 Let us compute (on the board) some of the following limits: sin(x) lim x 0 x, cos(x) lim x π π 2x, and 2 lim e x x 1. x 0 x 2
L H^opital s rule Example 10 The indeterminate / : when computing limits of functions like f(x) g(x), as x c (or x c + or x c ), we often find out that f(x) ± and g(x) ± and so we are in the presence of an indeterminate /. Some times such limits exist, some times they do not exist (that is why we call it indetermination). For instance, we have the following limits: ln(x) tan(5x) lim, lim x 0 + x 1 2 x ( π 2) tan(x), lim ln(x) x 0 + cot(x), lim 1 x x 0 + ln(x). Which of the previous limits exist? Only the first three!
L H^opital s rule Inderteminate ( / ) Suppose that when x c (x c +, or x c ), we obtain that f(x) ± and g(x) ±, but in the approach g (x) is never 0. If f (x) f(x) g (x) γ, then g(x) γ.
L H^opital s rule Example 11 Let us compute (on the board) some of the following limits: ln(x) tan(5x) lim, lim x 0 + x 1 2 x ( π 2) tan(x), We will compute only the third. lim ln(x) x 0 + cot(x), lim 1 x x 0 + ln(x).
L H^opital s rule Example 12 The indeterminate (0 ): Let us find lim x 0 + x ln(x) (found on the board). The indeterminate ( ): Let us find lim x ( π 2) (tan(x) sec(x)) (found on the board). The indeterminate (0 0 ): Let us find lim x 0 +xx (found on the board). The indeterminate (1 ): Find lim x 0 +(1+x)1 x (left as an exercise: use that e ln(y) = y as in the previous example).
L H^opital s rule Summary L H rule If f(x) lim x c g(x) is of the indeterminate form 0 0 then or ± ±, f and g are differentiable near c (except possibly at c), g and g are nonzero near c (except possibly at c), f lim (x) x c g (x) exists or is equal to ±, f(x) lim x c g(x) = lim f (x) x c g (x). The conclusion also holds if x as long as f and g are differentiable on some interval (a, ) on which g and g are nonzero. It also holds when x c +, x c, and x.
1 0 Indeterminate forms of limits Summary Indeterminate forms of limits 0 0 ± ± 0, 0 ( ) 0 0
Non indeterminate forms of limits Summary Non indeterminate forms of limits 1 0 0 In all of the previous cases the limit is 0! Non indeterminate forms of limits 0 + + 1 In all of the previous cases the limit is!
A final example Example 13 Compute lim x 3x+cos(x) x Solution: first we notice that the indetermination is of the form. Wrong solution! Applying L Hôpital s rule we have 3x +cos(x) 3 sin(x) lim = lim x x x 1 DNE! This solution is wrong because L Hôpital s rule does not apply to this case. Indeed, the application of such a rule would require the limit lim to exists, which is not the case. x 3 sin(x) 1
Mean value theorem (MVT) MVT Let f be a function that satisfies the following hypothesis: 1 f is continuous on the closed interval [a,b] 2 f is differentiable on the open interval (a,b) then there is at least a point c (a,b) such that f (c) = f(b) f(a) b a
Mean value theorem (MVT) remarks MVT Let f be a function that satisfies the following hypothesis: 1 f is continuous on the closed interval [a,b] 2 f is differentiable on the open interval (a,b) then there is at least a point c (a,b) such that f (c) = f(b) f(a) b a Note that the theorem does not tell us how to find the number c. It only tells us that such a number exists.
Mean value theorem (MVT) remarks MVT Let f be a function that satisfies the following hypothesis: 1 f is continuous on the closed interval [a,b] 2 f is differentiable on the open interval (a,b) then there is at least a point c (a,b) such that f (c) = f(b) f(a) b a Geometrical interpretation: see the board
Mean value theorem (MVT) remarks MVT Let f be a function that satisfies the following hypothesis: 1 f is continuous on the closed interval [a,b] 2 f is differentiable on the open interval (a,b) then there is at least a point c (a,b) such that f (c) = f(b) f(a) b a Physical interpretation: if I run continuously for an hour in High Park and I described my trajectory using a function s : [0,1] R, the average velocity will coincide with the instantaneous velocity at some point of my trajectory, i.e., there is a time t 0 [0,1] such that s (t 0 ) = s(1) s(0) 1 0 = s(1) s(0).