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Which function dominates as x leftarrow infinity? 0.03x3 and 50x2 0.03x3 50x2 which function dominates as x leftarrow? ln(x + 3) and x0.4 ln(x + 3) x0.4 Consider the graphs of f(x),g(x) and their tangent lines at x=4 shown belwo (not drawn to scale). The tangent line to f(x) is y = 2(x-4) and the tangent line to g(x) is y = -0.9(x-4). limx leftarrow 4 f(x)/g(x) = Consider the graphs of f(x),g(x) and their tangent lines at x=4 shown below (not drawn to scale). The tangent line to f(x) is y = 2(x-4) and the tangent line to g(x) is y = -0.9(x-4). limx leftarrow 4 f(x)/g(x) = Find the sign of lim x leftarrow af(x)/g(x) from the figure. negative positive zero Determine if L'Hopital's rule applies in each case. lim x leftarrow pi cos(7x)/x-pi yes no lim t leftarrow infinity (ln(t))2/t yes no lim y leftarrow 0 4y/y2 lim x leftarrow infinity e-x/1+ln(x) Determine if L'Hopital's rule aplies in each case. Lim theta leftarrow 0 arctan(theta)/7theta yes no lim x leftarrow 0+ cot(x)/ln(x) lim t leftarrow 0 t2 + 7t/cosh(t) - 1 yes no lim z leftarrow 0 e2z -1/ez yes no Find the limit. Use l'Hopital's rule if it applies. lim x leftarrow 0 e6x -1/sin x

Explanation / Answer

1) 0.03x^3

2) lim(x->infinity) x^0.4/ln(x+4) = lim(x-> infinity) 0.4*x^-0.6(1+x)

= lim(x->infinity) 0.4*(x^0.4 + x^-0.6) = infinity

Hence, x^0.4 dominates

3) lim(x->4) f'(x)/g'(x) = 2/-0.9 = -2.22

4) lim(x->4) f'(x)/g'(x) = 2/-0.9 = -2.22

5) since both the slopes are negative, limit is positive

6) a) no

b) yes

c) no

d) no

7) a) yes

b) no

c) yes

d) no

8) Yes applies

lim(x->0) (e^(6x)-1)/sinx

= lim(x->0) 6e^(6x)/cosx

= 6

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