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FUNCTIONS, LIMITS, AND CONTINUITY
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Note the analogy with real numbers, polynomials corresponding to integers, rational functions to rational numbers, and so on.
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TRANSCENDENTAL FUNCTIONS The following are sometimes called elementary transcendental functions. 1. 2. Exponential function: f x ax , a 6 0; 1. For properties, see Page 3.
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Logarithmic function: f x loga x, a 6 0; 1. This and the exponential function are inverse functions. If a e 2:71828 . . . ; called the natural base of logarithms, we write f x loge x ln x, called the natural logarithm of x. For properties, see Page 4. Trigonometric functions (Also called circular functions because of their geometric interpretation with respect to the unit circle): sin x; cos x; tan x sin x 1 1 1 cos x ; csc x ; sec x ; cot x cos x sin x cos x tan x sin x
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The variable x is generally expressed in radians ( radians 1808). For real values of x, sin x and cos x lie between 1 and 1 inclusive. The following are some properties of these functions: sin2 x cos2 x 1 1 tan2 x sec2 x 1 cot2 x csc2 x sin x sin x cos x cos x tan x tan x sin x y sin x cos y cos x sin y cos x y cos x cos y sin x sin y tan x tan y tan x y 1 tan x tan y 4.
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Inverse trigonometric functions. The following is a list of the inverse trigonometric functions and their principal values: a y sin 1 x; =2 @ y @ =2 b y cos 1 x; 0 @ y @  c y tan 1 x; =2 < y < =2 d y csc 1 x sin 1 1=x; =2 @ y @ =2 e y sec 1 x cos 1 1=x; 0 @ y @  f y cot 1 x =2 tan 1 x; 0 < y < 
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Hyperbolic functions are de ned in terms of exponential functions as follows. These functions may be interpreted geometrically, much as the trigonometric functions but with respect to the unit hyperbola. a b c ex e x 2 ex e x cosh x 2 sinh x ex e x tanh x cosh x ex e x sinh x d 1 2 sinh x ex e x 1 2 e sech x cosh x ex e x cosh x ex e x f coth x sinh x ex e x csch x
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The following are some properties of these functions: 1 tanh2 x sech2 x cosh2 x sinh2 x 1 sinh x y sinh x cosh y cosh x sinh y cosh x y cosh x cosh y sinh x sinh y tanh x tanh y tanh x y 1 tanh x tanh y coth2 x 1 csch2 x sinh x sinh x cosh x cosh x tanh x tanh x
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FUNCTIONS, LIMITS, AND CONTINUITY
6. Inverse hyperbolic functions. If x sinh y then y sinh 1 x is the inverse hyperbolic sine of x. The following list gives the principal values of the inverse hyperbolic functions in terms of natural logarithms and the domains for which they are real. p ! p 1 x2 1 1 1 2 1 ; all x d csch x ln a sinh x ln x x ; x 6 0 jxj x
b c
cosh
p x ln x x2 1 ; x A 1
  1 1 x ; jxj < 1 tanh 1 x ln 2 1 x
p ! 1 x2 e sech x ln ;0 < x @ 1 x   1 x 1 f coth 1 x ln ; jxj > 1 2 x 1
LIMITS OF FUNCTIONS Let f x be de ned and single-valued for all values of x near x x0 with the possible exception of x x0 itslef (i.e., in a deleted  neighborhood of x0 ). We say that the number l is the limit of f x as x approaches x0 and write lim f x l if for any positive number  (however small) we can nd some x!x0 positive number  (usually depending on ) such that j f x lj <  whenever 0 < jx x0 j < . In such case we also say that f x approaches l as x approaches x0 and write f x ! l as x ! x0 . In words, this means that we can make f x arbitrarily close to l by choosing x su ciently close to x0 .
 2  x if x 6 2 Let f x   0 if x 2 . Then as x gets closer to 2 (i.e., x approaches 2), f x gets closer to 4. We thus suspect that lim f x 4. To prove this we must see whether the above de nition of limit (with l 4) is EXAMPLE.
satis ed. For this proof see Problem 3.10. Note that lim f x 6 f 2 , i.e., the limit of f x as x ! 2 is not the same as the value of f x at x 2 since f 2 0 by de nition.