Limits (day 3)
Things we ll go over today 1. Limits of the form 0 0 (continued) 2. Limits of piecewise functions 3. Limits involving absolute values 4. Limits of compositions of functions 5. Limits similar to lim x 0 sin (x) x and lim x 0 1 cos (x) x 6. The Squeeze Theorem
1. Limits of the form 0 0 (continued)
1. Limits of the Form 0 0 If you are trying to find the limit of a fraction and the limit of the top and bottom are both 0, DO ALGEBRA!!! (for now) Some things to try Factoring Multiplying top and bottom by a conjugate Getting a common denominator
1. Limits of the Form 0 0 Ex 1 (Sec. 2.3, hw #24, 32, 8 th ed.): Find g) lim h 0 (3 + h) 1 3 1 h h) lim h 0 1 (x + h) 2 1 x 2 h
2. Limits of Piecewise Functions
2. Limits of Piecewise Functions Many functions are described using a single formula. A piecewise function is a function that needs more than one formula to be completely described. Ex: f x = 7 + x if x < 3 3x + 11 if 3 x 1 x 2 1 x 1 if x > 1
2. Limits of Piecewise Functions Recall: When evaluating piecewise functions, you must look at your input to decide which formula to plug it in to. Do not plug in your input to more than one formula!
2. Limits of Piecewise Functions Ex 2: For the function f defined by f x = 7 + x if x < 3 3x + 11 if 3 x 1 x 2 1 x 1 if x > 1 find a) f( 2) b) f(4) c) f( 5) d) f(1) e) f( 3)
2. Limits of Piecewise Functions Ex 3: For the function g defined by g x = sinx x if x 0 1 if x = 0 find a) g π 2 b) g(0)
2. Limits of Piecewise Functions Ex 4: For the function f defined by f x = 7 + x if x < 3 3x + 11 if 3 x 1 x 2 1 x 1 if x > 1 find a) lim x 4 f(x) b) lim x 3 f(x) c) lim x 1 f(x)
2. Limits of Piecewise Functions Ex 5: For the function g defined by g x = sinx x if x 0 1 if x = 0 find a) lim x π 4 g(x) b) lim x 0 g(x)
3. Limits Involving Absolute Values
3. Limits Involving Absolute Values Absolute values are piecewise functions!!! Def: x = Think of this as x if x 0 x if x < 0 inside = the inside if the inside is positive (or 0) and inside = the negative of the inside if the inside is negative
3. Limits Involving Absolute Values Ex 6: Calculate a) 8 b) 4 c) 12 d) 1 e) 2 8 f) e π g) ln (2) + 4 h) 1 10 i) 4 x if x < 4 j) 4 x if x > 4
3. Limits Involving Absolute Values Ex 7: Find lim x 7 x 1 x 1
3. Limits Involving Absolute Values Ex 8: Find lim x 1 x 1 x 1
3. Limits Involving Absolute Values Ex 9: Find lim x 2 x 2 4 2 x
3. Limits Involving Absolute Values Ex 10: Find lim x 0 1 x 1 x
4. Limits of Compositions of Functions
4. Limits of Compositions of Functions Def: Given functions f and g, the composition f composed with g (notation f g ) is defined by In words f g x = f(g x ) To find the formula for the composition of 2 functions, plug the formula for the function on the right (g in this case) into the formula for the function on the left (f in this case) in place of all the x s in the formula on the left (f in this case)
4. Limits of Compositions of Functions Ex 11: If f x = x2 + 5 2x 1 and g(x) = xcos(x) find a) f g x b) g f x c) f f x d) g g x
4. Limits of Compositions of Functions lim x a (f g)(x) Result: When calculating, 1. Calculate and see if it exists. If this limit exists, let s call it L. 2. Then lim x a g(x) lim x a f g x = lim x L f(x)
4. Limits of Compositions of Functions Note: This result can be extended to be useful for some additional problems by paying attention to lim x a g(x) and seeing if the answer is L + or L
4. Limits of Compositions of Functions Ex 12: Find a) lim x 3π 2 e sin (x) b) lim x 0 1 cos (x)
5. Limits Similar to lim x 0 sin (x) x and lim x 0 1 cos (x) x
5. Limits Similar to lim and x 0 sin (x) Two special limits to memorize x 1 cos (x) = 1 lim = 0 x 0 x lim x 0 sin (x) x = 1 lim x 0 1 cos (x) x = 0
5. Limits Similar to lim and Ex 13: Find x 0 sin (x) x 1 cos (x) = 1 lim = 0 x 0 x a) lim x 0 sin (5x) x b) lim x 0 sin (2x) 7x c) lim x 0 sin (4x) sin (9x) d) lim x 0 tan (3x) 5x
5. Limits Similar to lim and Ex 13: Find x 0 sin (x) x 1 cos (x) = 1 lim = 0 x 0 x e) lim x π 2 sin x π 2 x π 2 f) sin (cos 1 x ) lim x 1 cos 1 (x)
5. Limits Similar to lim and Ex 13: Find x 0 g) lim x 0 1 cos 4x 2x sin (x) x 1 cos (x) = 1 lim = 0 x 0 x
6. The Squeeze Theorem
Meaning 6. The Squeeze Theorem If f is any function
6. The Squeeze Theorem Meaning h is a function whose graph is always above or on the graph of f
Meaning 6. The Squeeze Theorem lim f x = L and lim h x = L x a x a
6. The Squeeze Theorem Meaning and g is a function whose graph is always between the graph of f and the graph of h
Meaning 6. The Squeeze Theorem then lim x a g x = L
Meaning If f is any function 6. The Squeeze Theorem h is a function whose graph is always above or on the graph of f lim f x = L and lim h x = L x a x a and g is a function whose graph is always between the graph of f and the graph of h then lim x a g x = L
Ex 14: If 12x 16 f(x) x 3 for all values of x near 2, find 6. The Squeeze Theorem lim x 2 f(x)
6. The Squeeze Theorem Ex 15: Find lim x 0 x2 sin 3 x x + 1
6. The Squeeze Theorem Ex 16: Find lim x 0 x3 cos 1 x + ln x ex
6. The Squeeze Theorem Ex 17: Find lim x 0 + xesin π x