what is the limit x-> -infinity x+ln(x^6+2)/ln(x^4+1)

Answer 1

The answer is -∞.

Note that [ x + ln (x^6+2)] / ln (x^4+1) is an indeterminate form with

x + ln (x^6+2) → -∞, since x → -∞ faster than ln (x^6+2) → +∞,

ln (x^4+1) → ∞,

as x → -∞.

So you can apply L'Hôpital's rule to get (after some algebraic manipulation):

[1 + 6x⁵ / (2 + x⁶)] • [x/4 + 1/(4x³ )].

Now note that

6x⁵ / (2 + x⁶)] → 0 and 1/(4x³ ) → 0 as as x → -∞,

so

[1 + 6x⁵ / (2 + x⁶)] → 1 and x/4 + 1/(4x³ ) → -∞.

Therefore, the product goes to -∞.

Again, please let me know if you have any questions.

Answer 2

Is the quantity like this x+ln/ln or is it like this (x+ln)/ln ?