Evaluate lim t rightarrow 0 sin4tt Evaluate lim x rightarrow

Evaluate: lim t rightarrow 0 sin4t/t Evaluate: lim x rightarrow 3 + 4x + 3x2/3 - x Find the derivatives by the limit process: g(x) = 3 - 4x2. Write the equation of the tangent line to graph of y = x + 4/x - 4 at the point (2, - 3). Do not simply Find the derivative: f(t) = + 6 - 4e2t

Solution

2) a) limt->0 sin4t/t = limt->0 (sin4t/4t)*4 = 4 b) limx->3+ (4x+3x^2)/(3-x) since numerator is finite & denominator is infinite hence limit = infinity 3) g(x) = 3-4x^2 g\'(x) = limh->0[f(x+h) - f(x)]/h g\'(x) = limh->0[3-4(x+h)^2 - 3+4x^2]/h g\'(x) = limh->0[3-4x^2 - 8hx - 4h^2 - 3 + 4x^2]/h g\'(x) = limh->0[-8hx-4h^2]/h g\'(x) = limh->0[-8x - 4h] g\'(x) = -8x 4) y = (x+4)/(x-4) y\' = [(x-4)d/dx(x+4) - (x+4)d/dx(x-4)]/(x-4)^2 y\' = -8/(x-4)^2 at (2,-3) y\' = -2 eq of tangent is y+3 = -2(x-2) y+3 = -2x+4 2x+y = 1 5) f(t) = t^1/4 + 6 - 4e^2t f\'(t) = 1/4t^-3/4 - 8e^2t