Question
Find the limit. (If the limit is infinite, enter ' \( \infty \) ' or ' \( -\infty \) ', as appropriate. If the limit does not otherwise exist, enter DN \[ \lim _{x \rightarrow-\infty}\left(x^{6}+2 x^{9}\right) \]
Ask by Patel Wagner. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limit is \( -\infty \).
Solution
To analyze the limit
limₓ→-∞ (x⁶ + 2x⁹),
notice that as x approaches -∞, the highest power term (2x⁹) dominates the behavior of the expression because x⁹ grows much faster than x⁶ in magnitude.
Since the exponent 9 is odd, when x → -∞, x⁹ → -∞. Multiplying by the positive coefficient 2 still results in -∞. Although x⁶ is always non-negative, it is insignificant compared to the dominating term 2x⁹.
Therefore, the limit is -∞.
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Simplify this solution Bonus Knowledge
To find the limit \(\lim _{x \rightarrow-\infty}\left(x^{6}+2 x^{9}\right)\), we can analyze the expression as \(x\) approaches \(-\infty\). The term \(2x^9\) dominates \(x^6\) when \(x\) is very large in the negative direction, since the exponent 9 is higher than 6. Hence, as \(x\) approaches \(-\infty\), \(2x^9\) will approach \(-\infty\) more rapidly than \(x^6\) can offset it. Thus, we can write: \[ \lim_{x \to -\infty} (x^6 + 2x^9) = \lim_{x \to -\infty} 2x^9 + x^6. \] Since \(2x^9\) approaches \(-\infty\), the whole limit is dominated by this term: \[ \lim_{x \to -\infty} (x^6 + 2x^9) = -\infty. \] Therefore, the limit is \(-\infty\).
