Compute [math: limx→−∞x7]\displaystyle \lim _{x o -\infty }{x^7}. (Use I to stand for [math: ∞]\infty if needed.)

Question

Compute [math: limx→−∞x7]\displaystyle \lim _{x	o -\infty }{x^7}. (Use I to stand for [math: ∞]\infty if needed.)

BlackTom AI · Accepted Answer

We are evaluating the limit as x approaches negative infinity of x^7.

First, recall that raising x to an odd power preserves the sign of x: if x is negative, x^7 remains negative, and as |x| grows larger, |x^7| grows without bound.

Second, consider the behavior as x decreases without bound (x → -∞). Since x is negative and its magnitude |x| increases without bound, x^7 = x · x · x · x · x · x · x will multiply negative numbers together seven times, resulting in increasingly large negative values.

Therefore, the function x^7 tends to negative infinity as x tends to negative infinity, because the magnitude grows without limit while the sign stays negative.

Note on notation: writing -∞ conveys the direction of the unbounded decrease. If one were to imagine infinity in the opposite direction, ∞ would imply unbounded growth in the positive direction, which does not describe this limit since the seventh power of a large negative x remains negative.

In summary, the limit diverges to negative infinity due to the odd power and the unbounded decrease of x.

_MATH1013_1ABCD_2025 Subsection 3.2 (closed on 4 Oct)

Compute [math: limx→−∞x7]\displaystyle \lim _{x\to -\infty }{x^7}. (Use I to stand for [math: ∞]\infty if needed.)

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