Let a = lim(x→0) (√1+√1+x⁴-√2)/x⁴ and b = lim(x→0) sin² x/(√2-√1+cos x). Find a*b³.

Let f(x) = a(7x-12-x²)/[b|x²-7x+12|] for x < 3, 2^{(sin(x-3)/(x-[x]))} for x > 3, and b at x = 3. Let S be the set of (a, b) making f continuous at 3. Find |S|.

Let f(x) = x³ + x²*f'(1) + x*f''(2) + f'''(3). Find f'(10).

If lim(x→0) (3 + a*sin x + b*cos x + log_e(1-x))/(3*tan² x) = 1/3, find 2a - b.

Let f(x) = x/2 + 2/x on (0, 2) and g(x) = min{f(t) : 0 < t ≤ x} for 0 < x ≤ 1, with a second branch for 1 < x < 2. Choose the correct statement about continuity and differentiability of g.

Let f:[1/2, 1] → R, f(x) = 4*√2*x³ - 3*√2*x - 1.
(I) y = f(x) meets the x-axis at exactly one point.
(II) It meets it at x = cos(pi/12).
Choose the correct statement(s).

Let f(x) = 1/|x| for |x| < 2 and a*x² + 2b for |x| ≥ 2. If f is differentiable on R, find 48(a + b).

Let g be non-constant and twice differentiable with g'(1/2) = g'(3/2), and f(x) = 1/2*[g(x) + g(2-x)]. Choose the correct statement about zeros of f'.

Find lim(x→0) (e^{(2|sin x|)} - 2|sin x| - 1)/x².

Let g be linear and f(x) = g(x) for x ≤ 0, ((1+x)/(2+x))^(1/x) for x > 0. If f is continuous at 0 and f'(1) = f(-1), find g(3).

Let f:(0, inf) → R, f(x) = e^{(-|log x|)}. Let m, n be the numbers of points where f is not continuous / not differentiable. Find m + n.

Let {x} denote the fractional part and f(x) = cos^-1(1-{x}²)*sin^-1(1-{x})/({x}-{x}³) for x ≠ 0. Let L, R be the left and right limits at 0. Find (32/pi²)(L² + R²).

Let f(x) = (a - b*cos 2x)/x² for x < 0, x² + c*x + 2 for 0 ≤ x ≤ 1, and 2x + 1 for x > 1. If f is continuous everywhere, and m is the number of points where f is not differentiable, find m + a + b + c.

Let f(x) = x - 1 if x is even, 2x if x is odd, x ∈ N. For some a ∈ N, f(f(f(a))) = 21. Find the left-hand limit at a involving |x|³/a and a greatest-integer term.

Let f(x) = |2x² + 5|x| - 3|. Let m, n be the numbers of points where f is not continuous / not differentiable. Find m + n.

If lim(x→1) ((5x+1)^(1/3)-(x+5)^(1/3))/((2x+3)^(1/2)-(x+4)^(1/2)) = m*√5/[n*(2n)^(2/3)] with gcd(m, n) = 1, find 8m + 12n.

Let f(x) = (1 - cos 2x)/x² for x < 0, a at x = 0, and b*√1-cos x/x for x > 0. If f is continuous at 0, find a² + b².

Let f(x) = (72^x - 9^x - 8^x + 1)/(√2-√1+cos x) for x ≠ 0, and a * log_e 2 * log_e 3 at x = 0. If f is continuous at 0, find a².

Let f(x) = x⁵ + 2x³ + 3x + 1 and g(f(x)) = x for all x in R. Find g(7)/g'(7).

Let f(x) = (sin 3x + a*sin x - b*cos 3x)/x³, continuous at 0. Find f(0).

Let f(x) = sin x + 3x - 2x² + x on (0, 2) (formula as rendered on the source page).
(I) f is increasing on (0, 2).
(II) f is decreasing on (0, 2).
Choose the correct statement(s).

Let a > 0 be a root of 2x² + x - 2 = 0. If lim(x→1/a) 16(1-cos(2+x-2x²))/(1-a*x²) = a + b*√17 with a, b integers, find a + b.

Let f:[-1, 2] → R, f(x) = 2x² + x + [x²] - [x]. Find the number of points where f is not continuous.

Let f:(-inf, inf)-{0} → R be differentiable with f'(1) = lim(a→∞) a²*f(1/a). Evaluate lim(a→∞) [a(a+1)/2 * tan^-1(1/a) + a² - 2*log_e a].

Let f(x) = x^x for x > 0. Find the interval on which f is strictly increasing.

Find lim(x→0) 2*(1 - cos x * √cos 2x * ∛cos 3x * ... * (10th root of cos 10x))/x².

Let f(x) = cos x - x + 1.
S1: f(x) = 0 for only one value of x in [0, pi].
S2: f is decreasing on [0, pi/2] and increasing on [pi/2, pi].
Choose the correct statement(s).

A circle passes through (2, 0) with centre (h, k). Let (x_c, y_c) be the intersection of 3x + 5y = 1 and (2+c)x + 5c² y = 1, with h = lim(c→1) x_c and k = lim(c→1) y_c. Find the circle equation.

Let f(x) = (8/7)^{(tan 8x/tan 7x)} for 0 < x < pi/2, a - 8 at x = pi/2, and (1+|cot x|)^{((b/a)|tan x|)} for pi/2 < x < pi, with a, b integers. If f is continuous at pi/2, find a² + b².

Find lim(x→0) (e - (1+2x)^(1/(2x)))/x.

Let f(x) = -3a² - 2 for x < 1 and a² + b*x for x ≥ 1. If f is differentiable on all R with a > 1, and the area enclosed by y = f(x) and y = -20 is a + b*√3, find a + b.

Let f be real differentiable with f(0) = 1 and f(x+y) = f(x)f'(y) + f'(x)f(y). Find sum(n=1..100) log_e f(n).

If lim(x→∞) ((e/(1-e))*(1/e - x/(1+x)))^x = a, find log_e a/(1 + log_e a).

Let f(x) = (2/x){sin((k1+1)x) + sin((k2-1)x)} for x < 0, 4 at x = 0, and (2/x)*log_e((2+k1*x)/(2+k2*x)) for x > 0. If f is continuous at 0, find k1² + k2².

The set of a for which 5x³ - 15x - a = 0 has three distinct real roots is (a, b). Find b - 2a.

Find lim(x→∞) (2x² - 3x + 5)(3x-1)^(x/2) / [(3x² + 5x + 4)*√(3x+2)^x].

A spherical chocolate ball has a uniform ice-cream layer. When the layer is 1 cm thick, the ice cream melts at 81 cm³/min and the thickness decreases at 1/(4*pi) cm/min. Find the surface area (in cm²) of the ball without the layer.

Find lim(x→0) cosec x * (√2*cos² x + 3*cos x - √cos² x + sin x + 4).

Let f:R-{0} → R satisfy f(x) - 6f(1/x) = 35/(3x) - 5/2. If lim(x→0) (1/(a*x) + f(x)) = b, find a + 2b.

Let f(x) = [x] + |x - 2| on -2 < x < 3, where [x] is the greatest integer. Let m, n be the numbers of points where f is not continuous / not differentiable. Find m + n.

Let f(x) = 3x for x < 0, min{1 + x + [x], x + 2[x]} for 0 ≤ x ≤ 2, and 5 for x > 2. Let a, b be the numbers of points where f is not continuous / not differentiable. Find a + b.

Let f(x) = lim(n→∞) sum(r=0..n) (tan(x/2^(r+1)) + tan³(x/2^(r+1)))/(1 - tan²(x/2^(r+1))). Find lim(x→0) (e^x - e^(f(x)))/(x - f(x)).

Let [t] denote the greatest integer. Find the least p in N such that lim(x→0+) (x*([1/x] + ... + [p/x]) - x²*([1/x²] + ... + [9²/x²])) ≥ 1.

If lim(t→0) (integral₀¹ (3x+5)^t dx)^(1/t) = a/(5e)*(8/5)^(2/3), find a.

Let f(x) = (x² - 1)|x² - a*x + 2| + cos|x|, not differentiable at x = a = 2 and x = b. Find the distance of (a, b) from 12x + 5y + 10 = 0.

Let a, b, g ∈ R. If lim(x→0) (x²*sin(a*x) + (g-1)e^(x2))/(sin 2x - b*x) = 3, find b + g - a.

Let f be twice differentiable with (sin x*cos y)(f(2x+2y) - f(2x-2y)) = (cos x*sin y)(f(2x+2y) + f(2x-2y)) and f'(0) = 1/2. Find 24*f''(5*pi/3).

If lim(x→0) (cos 2x + a*cos 4x - b)/x⁴ is finite, find a + b.

If lim(x→0) (tan x/x)^(1/x2) = p, find 96*log_e p.

If lim(x→1+) ((x-1)(6 + lambda*cos(x-1)) + mu*sin(1-x))/(x-1)³ = -1, find lambda + mu.

Let f(x) = max{x, x³, ..., x²¹}. Let m, n be the numbers of points where f is not differentiable / not continuous. Find m + n.

Let f be continuous with f(0) = 1 and f(2x) - f(x) = x. Let G(x) = lim(n→∞) {f(x) - f(x/2ⁿ)}. Find sum(r=1..10) G(r²).

Let f be differentiable with f(2) = 1 and f'(2) = 4. If lim(x→0) (f(2+x))^(3/x) = e^a, how many times does y = 4x³ - 4x² - 4(a-7)x - a meet the x-axis?

Find lim(x→0+) tan(5*x^(1/3))*log_e(1+3x²) / [(tan^-1(3*√x))² * (e^(5*x(4/3)) - 1)].

Let f(x) = [x²/2] - [√x] for x in [0, 4]. Find the number of points of discontinuity.

Let f(x) = (tan(tan x) - sin(sin x))/(tan x - sin x), continuous at 0. Find f(0).

Let t > -1, and let a_t, b_t be the roots of ((t+2)^(1/7) - 1)x² + ((t+2)^(1/10) - 1)x + ((t+2)^(2/…) - 1) = 0. If lim(t→-1) a_t = a and lim(t→-1) b_t = b, find 72(a + b)².

Statement I: lim(x→0) (tan^-1 x + log_e √(1+x)/(1-x) - 2x)/x⁵ = 2/5.
Statement II: lim(x→1) x^(2/(1-x)) = 1/e².
Choose the correct statement(s).

Let f be twice differentiable with f(3) = 18, f'(3) = 0, f''(3) = 4. Find lim(x→1) log_e((f(2+x)/f(3))^(18/(x-1)2)).

If lim(x→0) (e^((a-1)x) + 2*cos(b*x) + (c-2)e^(-x))/(x*cos x - log_e(1+x)) = 2, find a² + b² + c².

Let [.] denote the greatest integer, f(x) = min{√2*x, x²}, and S = {x in (-2, 2) : g(x) = |x|*[x²] is discontinuous at x}. Find sum(x in S) f(x).

Let f(x) = (a*x² + 2a*x + 3)/(4x² + 4x - 3) for x ≠ -3/2, 1/2, and b at x = -3/2, 1/2. If f is continuous at x = -3/2 and f o f(x) = 7/5, find x.

Let f(x) = (a|x| + x² - 2*sin|x|*cos|x|)/x for x ≠ 0, and b at x = 0. If f is continuous at 0, find a + b.

Let f(x) = 2a(x² - 2) + 2b*x for x < 1 and (a+3)x + (a-b) for x ≥ 1. If f is differentiable on all R, find 34(a + b).

Let f(x) = [e^x(e^{(tan x - x)} - 1) + log_e(sec x + tan x) - x*tan x - x]/(tan x - x), continuous at 0. Find f(0).

Let f(x) = b²*sin(pi/2*[(pi/2)(cos x + sin x)*cos x]) for x < 0, (sin x - 1/2*sin 2x)/x³ for x > 0, and a at x = 0. If f is continuous at 0, find a² + b².

Let y be differentiable on (0, inf) with y(1) = 2. If lim(t→x) (t²*y(x) - x²*y(t))/(x - t) = 3 for x > 0, find 2*y(2).

Find lim(x→0) log_e(sec(e^x)*sec(e^(2x))*...*sec(e^(10x))) / (e² - e^{(2*cos x)}).

If lim(x→2) sin(x³ - 5x² + a*x + b)/((√x-1 - 1)*log_e(x-1)) = m, find a + b + m.

Let f(x) = [x² - x - 1/2] on [2, 4]. Find the number of points of discontinuity.

Let f(x) = max{6x, 2 + 3x²} + |x - 1|*cos|x² - 1/4| for x in (-pi, pi). Find the number of points where f is not differentiable.

Let f(x) = e^x - 1 for x < 0 and x² - 5x + 6 for x ≥ 0, and g(x) = f(|x|) + |f(x)|. Let a, b be the numbers of points where g is not continuous / not differentiable. Find a + b.

Find the product of all a such that lim(x→0) (1 - cos(a*x)*cos((a+1)x)*cos((a+2)x))/sin²((a+1)x) = 2.

Let f(x) = lim(y→0) (1 - cos(x*y))*tan(x*y)/y³. Find the number of solutions of f(x) = sin x, x ∈ R.

Find lim(x→0) (x²*sin² x)/(x² - sin² x).

Let f(x) = x³ + 8 for x < 0 and x² - 4 for x ≥ 0, and g(x) = (x-8)^(1/3) for x < 0 and (x+4)^(1/2) for x ≥ 0. Find the number of points where g o f is discontinuous.

If lim(x→2) tan(x-2)*(r*x² + (p-2)x - 2p)/(x-2)² = 5 with r, p ∈ R, and the set of q such that the roots of r*x² - p*x + q = 0 lie in (0, 2) is (a, b], find 4(a + b).