exercise:
code show as below:
1. Limit[(Exp[x] - Exp[-x])/Sin[x], x -> 0] Limit[(x/(x - 1)) - 1/Log[x], x -> 1] Limit[(1 - Cos[x])/x^2, x -> 0] Limit[(1 + 1/x^2)^x, x -> Infinity] 2. D[a^x Log[x], x] D[(1 - Log[x])/(1 + Log[x]), x] D[(1 + (1 + x^(1/3))^(1/3))^(1/3), x] D[ArcTan[(1 + x)/(1 - x)], x] D[Exp[x] + Exp[Exp[x]], x] D[x^(x^x), x] D[(Sin[x])^Cos[x], x] D[Log[Cos[ArcTan[(Exp[x] - Exp[-x])/2]]], x] 3. D[Sin[x] Sin[2 x] Sin[3 x], {x, 20}] D[ArcTan[x], {x, 20}] D[1/(1 - x^2), {x, 60}] D[(1 + x)/Sqrt[1 - x], {x, 60}] 4. Integrate[(2 x - 3)^100, x] Integrate[(x + 1)/Sqrt[x], x] Integrate[x^2 a^x, x] Integrate[(2 x^2 - 5)/(x^4 - 5 x^2 + 6), x] Integrate[Log[x + Sqrt[1 + x^2]], x] Integrate[(Exp[2 x] + 1)/(Exp[x] + 1), x] Integrate[ArcTan[y, x], x, y] Integrate[x y z (1 - x - y), x, y, z] 5. Integrate[Sin[x]^2 Cos[x]^2, {x, 0, 1}] Integrate[Sqrt[Exp[x] - 1], {x, 0, Log[2]}] Integrate[Sqrt[Exp[x]]/Sqrt[Exp[x] + Exp[-x]], {x, 0, 1}] Integrate[x^2/Sqrt[x^2 + a^2], {x, 0, a}] Integrate[x^2 + y^3, {x, 1, 2}, {y, 1, 1 - x}] Integrate[x y^2, {x, 0, 1}, {y, x^2, x}] Integrate[r^2 Sin[y]^2, {y, 0, 2 Pi}, {r, 0, a}] Integrate[x y z, {x, 0, 1}, {y, 0, x}, {z, 0, x + y}] 6. Series[Exp[x^2], {x, 0, 5}] Series[x^2/(1 - x), {x, 0, 5}] Series[Log[Sqrt[(1 + x)/(1 - x)]], {x, 0, 5}] Series[(1 + x) Exp[-x], {x, 0, 5}] Series[Cos[x] Cos[y], {x, 0, 5}, {y, 0, 5}] 7. DSolve[{Derivative[1][y][x] == x + y[x], y[0] == 1}, y[x], x] DSolve[{x'[t] + y[t] == Cos[t], y'[t] + x[t] == Sin[t]}, {y[t], x[t]}, t] DSolve[{x'[t] == 2 x[t] - y[t] + z[t], y'[t] == 2 x[t] + 2 y[t] - z[t], z'[t] == x[t] + 2 y[t] - z[t]}, {x[t], y[t], z[t]}, t]