Calculus cheat sheet all reduced

Calculus Cheat Sheet

Calculus Cheat Sheet

Limits
Definitions Limit at Infinity : We say lim f ( x ) = L if we Precise Definition : We say lim f ( x ) = L if
x®a x®¥

for every e > 0 there is a d > 0 such that whenever 0 < x - a < d then f ( x ) - L < e . “Working” Definition : We say lim f ( x ) = L if we can make f ( x ) as close to L as we want by taking x sufficiently close to a (oneither side of a) without letting x = a . Right hand limit : lim+ f ( x ) = L . This has
x® a x®a

can make f ( x ) as close to L as we want by taking x large enough and positive. There is a similar definition for lim f ( x ) = L
x® - ¥

Evaluation Techniques L’Hospital’s Rule Continuous Functions f ( x) 0 f ( x) ± ¥ If f ( x ) is continuous at a then lim f ( x ) = f ( a ) x®a If lim = orlim = then, x® a g ( x ) x®a g ( x ) ±¥ 0 Continuous Functions and Composition f ( x) f ¢( x) lim = lim a is a number, ¥ or -¥ f ( x ) is continuous at b and lim g ( x ) = b then x® a g ( x ) x®a g ¢ ( x ) lim f ( g ( x ) ) = f lim g ( x ) = f ( b )
x®a x ®a

except we require x large and negative. Infinite Limit : We say lim f ( x ) = ¥ if we can make f ( x ) arbitrarily large (and positive) bytaking x sufficiently close to a (on either side of a) without letting x = a . There is a similar definition for lim f ( x ) = -¥ except we make f ( x ) arbitrarily large and negative.
x®a x®a

(

)

x®a

Factor and Cancel ( x - 2 )( x + 6 ) x 2 + 4 x - 12 = lim lim x®2 x ®2 x2 - 2 x x ( x - 2) x+6 8 = =4 x®2 x 2 Rationalize Numerator/Denominator 3- x 3- x 3+ x lim 2 = lim 2 x ® 9 x - 81x ® 9 x - 81 3+ x 9-x -1 = lim = lim x®9 ( x 2 - 81) 3 + x x ®9 ( x + 9) 3 + x = lim

Polynomials at Infinity p ( x ) and q ( x ) are polynomials. To compute
x® ± ¥

lim

q ( x)

p (x)

factor largest power of x in q ( x ) out

of both p ( x ) and q ( x ) then compute limit. x 2 3 - 42 3 - 42 3x 2 - 4 3 x = lim 2 5 = lim 5 x = x® - ¥ 5x - 2 x 2 x® - ¥ 2 x ( x - 2) x® - ¥ x - 2 limPiecewise Function ì x 2 + 5 if x < -2 lim g ( x ) where g ( x ) = í î1 - 3 x if x ³ -2 Compute two one sided limits, lim- g ( x ) = lim- x 2 + 5 = 9
x ® -2 x ® -2 x ® -2

the same definition as the limit except it requires x > a . Left hand limit : lim- f ( x ) = L . This has the
x®a

(

)

same definition as the limit except it requires x 0 and sgn ( a ) = -1 if a < 0 . 1. 2.
x® ¥lim e x = ¥ & lim ln ( x ) = ¥

x® - ¥

lim e x = 0
x®0

5. n even : lim x n = ¥
x® ± ¥

x® ¥

&

lim+ ln ( x ) = - ¥

6. n odd : lim x n = ¥ & lim x n = -¥
x® ¥ x®- ¥

Some Continuous Functions Partial list of continuous functions and the values of x for which they are continuous. 1. Polynomials for all x. 7. cos ( x ) and sin ( x ) for all x. 2. Rational function, except for x’sthat give 8. tan ( x ) and sec ( x ) provided division by zero. 3p p p 3p 3. n x (n odd) for all x. x ¹ L, , - , , ,L 2 2 2 2 4. n x (n even) for all x ³ 0 . 9. cot ( x ) and csc ( x ) provided 5. e x for all x. x ¹ L , -2p , -p , 0, p , 2p ,L 6. ln x for x > 0 . Intermediate Value Theorem Suppose that f ( x ) is continuous on [a, b] and let M be any number between f ( a ) and f ( b ) . Thenthere exists a number c such that a < c < b and f ( c ) = M .

b 3. If r > 0 then lim r = 0 x® ¥ x 4. If r > 0 and x r is real for negative x b then lim r = 0 x® - ¥ x

7. n even : lim a x n + L + b x + c = sgn ( a ) ¥
x® ± ¥

8. n odd : lim a x n + L + b x + c = sgn ( a ) ¥ 9. n odd : lim a x + L + c x + d = - sgn ( a ) ¥
n

x® ¥

x ® -¥

Visit http://tutorial.math.lamar.edu for acomplete set of Calculus notes.

© 2005 Paul Dawkins

Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.

© 2005 Paul Dawkins

Calculus Cheat Sheet

Calculus Cheat Sheet

Derivatives
Definition and Notation f ( x + h) - f ( x ) . If y = f ( x ) then the derivative is defined to be f ¢ ( x ) = lim h®0 h If y = f ( x ) then all of the following are equivalent...