1. State the informal definition of the limit of a function. 4. What are the values of the following limits: lim. 2x 1 if x 1 discontinuous?

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1 Review for Limits Key Concepts 1 State the informal definition of the limit of a function 2 State 9 limit theorems 3 State the squeeze theorem 4 What are the values of the following limits: lim sin x x 5 Define the continuity of a function 6 State the limit composition theorem 7 State the intermediate value theorem cos and lim? x Practice Problems 2x +1 ifx 1 1 Where is g(x) = 3x if 1 <x<1 2 if x 1 discontinuous? (a) 0 only (b) 1 only (c) 1 & 1 only (d) no point of the domain (e) every point of the domain t if t<0 2 Where is v(t) = 1 if 0 <t 1 discontinuous? t if t>1 (a) 0 only (b) 1 only (c) 1 & 1 only 3 Find lim f(x), given f(x) = (d) no point of R (e) every point of the domain x if x<0 4 x if 0 x<4 (x 4) 2 if x>4

2 (a) 0 (b) 1/2 (c) 1 (d) does not exist (e) not enough information given 4 If p(s) = 4 s, then what is the value of lim s 16 p(s)? s 16 (a) (b) 1 8 (c) 0 (d) 1 8 (e) 5 If s(v) = v2 +2v 8 v 4 16 (a) (b), then what is the value of lim s(v)? v Find lim f(x), given f(x) = (c) 1 (d) 3 2 x if x<0 4 x if 0 x<4 (x 4) 2 if x>4 (e) (a) 0 (b) 1/2 (c) 1 (d) does not exist (e) not enough information given 7 If u(t) = t 7,thenfind lim t 7 u(t) t 7 (a) (b) 1 (c) 0 (d) 1 (e) 7 8 Find lim x 3 x 3 x 2 2x 3 (a) Find lim f(x), given f(x) = (b) 1 3 (c) 0 (d) 3 3x +1 2 if x<1 0 if x =1 x 2 +2x 3 x 2 + x 2 if x>1 (e) does not exist

3 (a) does not exist (c) 0 (e) 4 3 (b) 3 (d) 3 4 3x + k x 2 10 Find a constant k such that f(x) = is continuous for 2x x > 2 all x R In particular, using your value of k, justifywhyf is continuous on R x 2 + a if x<0 11 Let a and b be constants and define f(x) = x 2 + b if x 0 (a) How should a and b be related to each other in order for f to be a continuous function on R? (b) How should the values of a and b be restricted in order for f to be a continuous function on R? 2x if x 1 12 Find lim f(x), if it exists, where f is defined by f(x) = 2 if x>1 sin(x 2) 13 Find lim, if it exists; explain x 2 14 Evaluate the following two limits, lim x cos 1 x and lim x 2 cos 1 x 2sinx x<0 x 15 Let f(x) = a x =0 Find the values of a and b so that f is b cos x x > 0 continuous for all real numbers? ax +3 x 2 16 Find a so that the function f(x) = satisfies lim f(x) =1 3 x x > 2 17 Explain why lim x 2 x 2 18 Determine a so that lim x π/4 19 Determine a so that lim does not exist a a tan x =1 sin x cos x x ax 2 =1

4 1 cos ax 20 Determine a so that lim =1 x 3 21 Evaluate lim and lim lim a and explain 22 Find the one-sided limit algebraically, lim h Find the one-sided limit algebraically, lim + 24 Find the two-sided limit, lim 6x 2 (cot x)(csc 2x) 25 Find the two-sided limit, lim h 0 26 Let f(x) = 2 if x<2 c +2 if x =2 cx +1 if x>2 Choose an interval that contains c 27 Find lim sin 1 x sin(sin h) 4 Conjecture a formula for sin h h2 +4h +5 5 h 2x() Determine c so that f is continuous 28 If f and g are continuous functions with f(3) = 5 and lim [2f(x) g(x)] = 4, x 3 find g(3) 29 Find lim x 5 2 x 5 f(x + h) f(x) 30 Find lim h 0 h 31 Find lim x 3 f(x) iff(x) = where f(x) =x 2 + x 2(x + 1) if x<3 4 if x =3 x 2 1 if x>3 32 Find lim cos 1 x 33 The function f(x) = x 3 x 3 is discontinuous where?

5 2x +1 if x 1 34 Let g(x) = 3x if 1 <x<1 2 if x 1 Where is g discontinuous at? ax 6 x = 2 35 Let f(x) = x 2 Find constants a and b are chosen such that b x =2 f is continuous for all x R The product ab is? 36 Find lim sin 2 ax bx find the value of k such that f is con- 3 x x = 9 37 Given f(x) = 9 x k x =9, tinuous on R where a and b are nonzero constants x a 38 Find lim, where a is a positive constant x 4 x a2 5x + k x 2 39 Find a constant k such that f(x) = is continuous for 2x x > 2, all real numbers x In particular, using your value of k, justify why f is continuous on R x 2 +2 x 1 40 If f(x) = then, at x =1,fis? x +2 x>1, ax Find a value for a so that lim =1 x 42 Find lim x +2 2 x 2, if it exists; explain 43 Let a and b denote constants and let f(x) = 4x + b if x<0 a 9 if x =0 Find sin ax if x>0 x values for a and b so that f is continuous on R Give a coherent mathematical argument to justify your values for a and b

6 Quiz 1 x Find lim x 2 x 2 algebraically 2 Find the average rate of change of the function g(x) =x 2 over the interval [ 1, 1] 3 Write a theorem which states when a two-sided limit exists in terms of one-sided limits 4 If f(1) = 5, must lim f(x) exist?ifitdoes,thenmust lim f(x) = 5? Quiz 2 1 (a) State the Sum Rule for limits (b) State the Squeeze Rule or (Sandwich Theorem) for limits 2 Let F (x) = x 2 +3x +2 (2 x ) find lim F (x) algebraically x 2 3 Suppose lim f(x) = 0 and lim g(x) = 3 Find, lim (g(x) + 3), lim xf(x), x 4 x 4 x 4 x 4 g(x) lim x 4 (g(x))2, and lim x 4 f(x) 1 4 Evaluate lim h 0 f(x + h) f(x) h Quiz 3 given f(x) =3x 4 and x =2 1 (a) State the Rational Functions Rule for limits (b) State the Squeeze Rule or (Sandwich Theorem) for limits 2 Determine the value of the limit given: lim x 5 x 5 x Determine the value of the limit given: lim x +3 2 f(x) 5 4 (a) If lim x 2 lim f(x) = 3, find lim f(x) (b) If lim f(x) 5 x 2 = 4, find