(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. Suppose that L1 = lim(x->a+) f1(x) and L2 = lim(x->a+) f2(x)

Also, suppose that f1(x)<=f2(x) for all x in (a,b). Show that L1<=L2

2. Suppose f(x) = (sqrt(1+3*x^2) - 1)/(x^2)

show that the lim (x->0) f(x) exits and give its value.

2. Relevant equations

3. The attempt at a solution

1) I find this problem a tad cumbersome for I have no clue as to where to start. I drew a graph which led me to the following conclusion

for any sequence x_n in (a,b)

1 [lim(n) f1(x_n) = f1(x_o)] <= [lim(n) f2(x_n) = f2(x_o)]

2 and maybe |L2 - L1| = |f2(x) - f1(x)|

but 1 shows that when x_n gets arbitrarily close to a, f1(a)<=f2(a)

2) I multiplied the top of f(x) by (sqrt(1+3*x^2) + 1). that is

(sqrt(1+3*x^2) - 1)/(x^2) * (sqrt(1+3*x^2) + 1) = 3.

hence the lim(x->0) f(x) = 3

thanks!!

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# Homework Help: Real analysis limits

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