Calculate the limit values: a) lim     x → ∞ ln (x ^ 2 + 2) / x   b) lim           x → 0 tan x / x c) lim       x → 0 e 2x - 1 - 2x / x 2 2 d) lim         x → ∞? (√x ^ 2 + 2x - x)

A) This particular limit is of the indeterminate form,

if we plug in infinity directly, though it is not a number just to check.

If a limit is in this form, we apply L'Hopital's Rule.

's

So we take the derivatives and obtain,



Still it is of the same indeterminate form, so we apply the rule again,



This simplifies to,



b) This limit is also of the indeterminate form,


we still apply the L'Hopital's Rule,





When we plug in zero now we obtain,


c) This also in the same indeterminate form





It is still of that indeterminate form so we apply the rule again, to obtain;



Now we have remove the discontinuity, we can evaluate the limit now, plugging in zero to obtain;



This gives us;



d)

For this kind of question we need to rationalize the radical function, to obtain;



We now divide both the numerator and denominator by x, to obtain,



This simplifies to,