Calculus Answers

Prove that : lim n→∞

(2n³+5n)/(4n³+n²) = 1/2

Let be any positive integer. Use sandwich theorem for sequences to prove that the sequence { 1 n} converges to 0.

Give an upper bound and a lower bound for the expression 1/(a^(4)+3a^(2)+1) if a∈R

Use the definition of limit to prove that the sequence {n − 1 }∞n=1 is divergent.

Use the definition of limit to prove that the sequence { 1 √ n+1} converges to 0.

Use the definition of limit to prove that both of the sequences { 1 √ n √ } and {(−1)n n } converges to 0.

Let {an}

∞

n=1 be a convergent sequence with limit L. Then use the definition of

limit to prove that

lim

n→∞

−an = −L.

Now if L = 0, then use the definition of limit to prove that

limn→∞

(−1)n

an = 0

Deduce that f(x)=x^(3)+3x^(2)-1 has exactly one zero in each of the intervals [0,1],[-1,0] and [-3,-2]

Show that f(x)=(x^3)+(3x^2)-1 has at least one zero in each of the intervals [0,1],[-1,0] and [-3,-2]

3.A quantity with an initial value of 980 grows exponentially at a rate of 9.5% every 7 years. What is the value of the quantity after 93 years, to the nearest hundredth?