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practice1 - 49 Worksheet 2.4 Limits and Continuity 1 Using...

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Unformatted text preview: 49 Worksheet 2.4. Limits and Continuity 1. Using a graphing calculator, graph f(:1:) = 81:31:. Show f has a removable discontinuity mx=0 2. Graph f (ac) = Show f has a jump discontinuity at x = 0. (I? 50 2 1 3. Graph f = Show I has an inﬁnite discontinuity at x = 0. a: 4. Determine where the following flinctions are discontinuous and classify the type of dis- continuity. a. mac) = iéxi b. f(t) = 3r.” — 9t3 ‘ 2_ c.f(x)=:_§ d. f(x)=::§ e. f(t) = tan 2t 64 Worksheet 2.8. The Formal Deﬁnition of a Limit . Suppose f (3:) = 3x — 5. How close must as be to 2 to ensure that the error | f (x) — 1| is less than 104? - . Find a number 6 such that [352 — 4| is less than 10’4 if 0 < In: —— 2| < 6. 1 . Prove rigorously that lim msin — = 0. 1—0 (I: . . 1 1 . Prove rigorously that 11m —— = —. m—r2 1:2 4 :1: . . Using the negation of the deﬁnition of the limit, prove that ling ﬂ does not ex18t. 1% In 1 . Using the negation of the deﬁnition of the limit, prove that 111% sin —— does not exist. 11—0 {B 61 Worksheet 2.7. Intermediate Value Theorem 1. Show that g(x) = x: 1 takes on the value 0.599 for some (t 6 [1,2]. 2. Show that cos 9 = 0 has a solution in the interval. [0, 1] 3. Using the Intermediate Value Theorem, show that f (x) = 3:3 — 8:1; — 1 has a root in the interval [2.75, 3]. Apply the Bisection Method twice to ﬁnd an interval of length 1% containing this root. 1 4. Suppose that f(x) = x+2 if x < —2 and that f(:1:) = §x+3 if x _>_ —2. Show that there does not exist a number c such that f (c) = l. 58 Worksheet 2.6. ’I‘rigonometric Limits 1 . Use the Squeeze Theorem to evaluate linrflJ 3:2 sin —. 22—) :L' 7T ). . Use the Squeeze Theorem to evaluate lini(x — 1) sin( 1 1—9 x —— sin(5h) . lim = . lim h—bO 3h \$2 w—vo sin2 ac , cost — coszt . 11m ————-——— = t—bD t . lim 1imar,'2—64__ z—*8 (II-8 — 1, m3—4x_ \$13; 33—2 .- . (h+2)2—9h_ £13}; h—4 ‘ . \/—-4_ JLIIIIBx—lﬁ— , cott .11m——-—-= tagcsct . linr}(sect — tant) = NI cos 9 — 1 0—»0 sin 6 Worksheet 2.5. Evaluating Limits Algebraically 55 ...
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