. Differential and integral calculus, an introductory course for colleges and engineering schools. o, and write 7T lim tan x = oo, which is further abbreviated to tan - = oo . This .IT ^ Z = 2 last expression is merely an abbreviation for the entirely accuratestatement, as x = -t tana; increases (numerically) without limit. q Similar explanations apply to such expressions as tan —- = oo,cot 0 = oo , sec - = oo, and esc 0 = oo . There is an ambiguity in § 18 LIMITS 27 these expressions, because, for example, lim tan x = + oo or — oo according as x = - from values less than ~ or greater than - •
. Differential and integral calculus, an introductory course for colleges and engineering schools. o, and write 7T lim tan x = oo, which is further abbreviated to tan - = oo . This .IT ^ Z = 2 last expression is merely an abbreviation for the entirely accuratestatement, as x = -t tana; increases (numerically) without limit. q Similar explanations apply to such expressions as tan —- = oo,cot 0 = oo , sec - = oo, and esc 0 = oo . There is an ambiguity in § 18 LIMITS 27 these expressions, because, for example, lim tan x = + oo or — oo according as x = - from values less than ~ or greater than - • It ispurely an arbitrary convention that we write tan ~ = + <» ratherthan — oo . Similar remarks apply in the other cases.* 18. Example of a Function That Has No Limit. The function TV sin - has no limit when x = 0. vx TT For, in that case, - increases x 7T without limit, while sin- oscil- ox lates between the extreme val-ues + 1 and — 1, passing through0 at each oscillation, but ap-proaching no limit. To con- 7 T~struct the graph of the function, we construct a table of values. of x and sin x as follows: x = 4, 4 2 4 43 4 45 46 4 7 4 _, . . 7T 7T x ~ 4 7T — > 2 3tt t TT 4 * 5tt4 3tt T 7?r,4 2x, . sin- = jV2, 1, JV2, 0, -|V2, -1, -|V2, 0. . x As a: continues to approach 0, this series of values of sin- is re- x peated again and again, showing that the function has no limit. * In the text the inaccuracy of giving literal interpretations to such ex-pressions as ^. = oo, tan - = oo, is dwelt upon in order to bring out more clearly the nature of a limit, and to teach the student to think accurately,and not because there is much danger of inaccurate results arising from theliteral interpretation of these expressions. Errors seldom arise in trigono-metric work from the literal interpretation of tan - = oo, sec 5 = 00, etc. 28 . DIFFERENTIAL CALCULUS §§19-20 In the figure only the right-hand branch is shown. A symmetricalbranch lies at the left of the
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