using Plots
f(k) = √(k+1) - √(k-1)
g(k) = 2/(√(k+1) + √(k-1))
h(k) = 1/√k
k = (10^9-1000):10^9
plot(k, f; label="√(k+1) - √(k-1)", alpha=0.5, lw=0.7)
plot!(k, g; label="2/(√(k+1) + √(k-1))", ls=:dash)
plot!(k, h; label="1/√k", ls=:dashdot)
plot!(rightmargin=5Plots.mm)
k = 10^9
@eval @show f($k) g($k) h($k);
f(1000000000) = 3.162277789670043e-5 g(1000000000) = 3.1622776601683795e-5 h(1000000000) = 3.1622776601683795e-5
k = big(10^9)
@eval @show f($k) g($k) h($k);
f(1000000000) = 3.162277660168379332394178251953765950392353891919764888393048074577188547896907e-05 g(1000000000) = 3.162277660168379332394178251953765950392353891919764888393048074577180873944593e-05 h(1000000000) = 3.162277660168379331998893544432718533719555139325216826857504852792594438639224e-05
using Plots
F(a, b, c) = (-b + √(b^2 - 4a*c))/(2a)
G(a, b, c) = (2c)/(-b - √(b^2 - 4a*c))
b, c = 1, 1
a = range(-3e-8, 3e-8, 2001)
plot(a, a -> F(a, b, c); label="(-b + √(b^2 - 4a*c))/(2a)", lw=0.7)
plot!(a, a -> G(a, b, c); label="(2c)/(-b - √(b^2 - 4a*c))", ls=:dash)
plot!(rightmargin=4Plots.mm)
n = 10^9
@eval @show F(1/$n, 1, 1) G(1/$n, 1, 1);
F(1 / 1000000000, 1, 1) = -1.0000000272292198 G(1 / 1000000000, 1, 1) = -1.000000001
n = big(10^9)
@eval @show F(1/$n, 1, 1) G(1/$n, 1, 1);
F(1 / 1000000000, 1, 1) = -1.000000001000000002000000005000000014000000042000000132000000429000000655609063 G(1 / 1000000000, 1, 1) = -1.000000001000000002000000005000000014000000042000000132000000429000001429999996