Limits and continuityChapter 3: Practice/review problemsThe collection of problems listed below contains questions taken from previous MA123 exams.Limits and one-sided limits[1]. Suppose H(t) = t2+ 5t + 1. Find the limit limt→2H(t).(a) 15 (b) 1 (c) 9 (d) 6 (e) 2t + 5[2]. Find the limit limt→2t2− 4t − 2.(a) 2 (b) 4 (c) 6 (d) 8 (e) The limit does not exist[3]. Find the limit limx→5x − 5x2− 25.(a) −110(b) −15(c) 0(d)15(e)110[4]. Compute limx→3x2− 7x + 12x − 3.(a) 0 (b) 1 (c) −1 (d) 2 (e) The limit does not exist[5]. Find limr→1r2− 3r + 2r − 1.(a) 1 (b) 0 (c) −1 (d) 2 (e) The limit does not exist[6]. Find the limit or state that it d oes not exist: limx→4x2+ x − 20x − 4.(a) 8 (b) −20 (c) −15 (d) 9 (e) Does Not Exist[7]. Compute limx→02x2− 3x + 4x+5x − 4x.(a) 5 (b) 4 (c) 3 (d) 2 (e) 1[8]. Compute limh→0(h + 4)2− 16h.(a) 4 (b) 5 (c) 6 (d) 7 (e) 834[9]. Find the limit limt→0+√t3√t.(a) 0 (b) 1 (c) 2 (d) 3 (e) The limit does not exist[10]. Find the limit as x tends to 0 from the left limx→0−|x|2x.(a) 1/3 (b) 1/2 (c) 0 (d) −1/2 (e) −1/3[11]. Find the limit limh→0−|4h|h.(Hint: Evaluate the quotient for some negative values of h close to 0.)(a) 0 (b) 2 (c) −2 (d) 4 (e) −4[12]. Compute limx→3−|4x − 12|x − 3.(a) 4 (b) −4 (c) 0 (d) Do esn’t exist (e) Cannot be d etermined[13]. Find the limit of f (x) as x tends to 2 from the left if f(x) =1 + x2if x < 2x3if x ≥ 2(a) 5 (b) 6 (c) 7 (d) 8 (e) 9[14]. Find the limit of f (x) as x tends to 2 from the left if f(x) =x3− 2 if x ≥ 21 + x2if x < 2(a) 5 (b) 6 (c) 7 (d) 8 (e) Does not exist[15]. For the function f(x) =4x2− 1 if x < 13x + 2 if x ≥ 1Find limx→1+f(x).(a) 5 (b) 3 (c) 1 (d) 0 (e) The limit does not exist[16]. Let f(x) =(x2+ 8x + 15 if x ≤ 24x + 7 if x > 2.Find limx→2+f(x).(a) 15 (b) 20 (c) 30 (d) 35 (e) The limit does not exist35[17]. Let f(x) =(−5x + 7 if x < 3x2− 16 if x ≥ 3.Find limx→3+f(x).(a) 6 (b) −6 (c) −7 (d) −8 (e) The limit does not exist[18]. Suppose f(t) =−t if t < 1t2if t ≥ 1Find the limit limt→1f(t).(a) −1 (b) 1 (c) 0 (d) 2 (e) The limit does not exist[19]. Suppose f(t) =(−t)2if t < 1t3if t ≥ 1Find the limit limt→1f(t).(a) −2 (b) −1 (c) 1 (d) 2 (e) The limit does not exist[20]. Suppose the total cost, C(q), of producing a quantity q of a product equals a fixed cost of $1000 plus $3times the quantity pro duced. So total cost in dollars isC(q) = 1000 + 3q.The average cost per unit quantity, A(q), equals the total cost, C(q), divided by the quantity produced,q. Find the limiting value of the average cost per unit as q tends to 0 from the right. In other words findlimq→0+A(q)(a) 0 (b) 3 (c) 1000 (d) 1003 (e) The limit does not existLimits at infinity[21]. Find the limit limt→∞31 + t2.(a) 0 (b) 1 (c) 2 (d) 3 (e) The limit does not exist[22]. Find the limit limx→∞x2+ x + 1(3x + 2)2.(a) 1 (b) 1/3 (c) 0 (d) 1/9 (e) The limit does not exist[23]. Find the limit lims→∞s4+ s2+ 13s3+ 8s + 9.(a) 0 (b) 1 (c) 2 (d) 3 (e) The limit does not exist36[24]. Find the limit limx→∞2x2(x + 2)3.(a) 0 (b) 1 (c) 2 (d) 3 (e) The limit does not exist[25]. Suppose the total cost, C(q), of producing a quantity q of a product is given by the equationC(q) = 5000 + 5q.The average cost per unit quantity, A(q), equals the total cost, C(q), divided by the quantity produced,q. Find the limiting value of the average cost per unit as q tends to ∞. In other words findlimq→∞A(q)(a) 5 (b) 6 (c) 5000 (d) 5006 (e) The limit does not existContinuity and differentiability[26]. Suppose f(t) =Bt if t ≤ 35 if t > 3Find a value of B such that th e function f(t) is continuous f or all t.(a) 3/5 (b) 4/5 (c) 5/3 (d) 5/4 (e) 5/2[27]. Suppose that f(x) =A + x if x < 21 + x2if x ≥ 2Find a value of A such th at the function f(x) is continuous at the point x = 2.(a) A = 8 (b) A = 1 (c) A = 2 (d) A = 3 (e) A = 0[28]. Suppose f(t) =(t if t ≤ 3A +t2if t > 3Find a value of A such th at the function f(t) is continuous for all t.(a) 1/2 (b) 1 (c) 3/2 (d) 2 (e) 5/2[29]. Consider the function f (x) =(2x2+ 3 if x ≤ 33x + B if x > 3.Find a value of B such that f (x) is continuous at x = 3.(a) 6 (b) 9 (c) 12 (d) 15 (e) There is n o such value of B.[30]. Find all values of a such that the function f(x) =x2+ 2x if x < a−1 if x ≥ ais continuous everywhere.(a) a = −1 only (b) a = −2 only (c) a = −1 and a = 1(d) a = −2 and a = 2 (e) all real numbers37[31]. Which of the following is true for the function f(x) given byf(x) =2x − 1 if x < −1x2+ 1 if −1 ≤ x ≤ 1x + 1 if x > 1(a) f is continuous everywhere(b) f is continuous everyw here except at x = −1 and x = 1(c) f is continuous everywhere except at x = −1(d) f is continuous everyw here except at x = 1(e) None of the above[32]. Which of the following is true for the function f(x) = |x − 1|?(a) f is differentiable at x = 1 and x = 2.(b) f is differentiable at x = 1, but not at x = 2.(c) f is differentiable at x = 2, but not at x = 1.(d) f is not differentiable at either x = 1 or x = 2.(e) None of the