Determine the following limits. lim x→−∞ (3x 7 + x 2 )

Welcome back everyone. In this problem, we want to find the limit as X approaches negative infinity of 7 X 15th plus X to the 4th. A says it's 7, B 8 C negative infinity, and the D says it's infinity. Now what do we know that can help us to figure out the limit as x approaches negative infinity? Well, first note here that this is a polynomial function, and the limit of the polynomial function is determined by the behavior of the leading term. In this case, our leading term is 7 x 15th. OK, so this the limit as X approaches negative infinity of this expression is going to be equal to the limit. As x approaches negative infinity of 7 X 15th. Now we can take out or or break out or we're constant here and rewrite this as 7 multiplied by the limit or as X approaches infinity. Sorry, X approaches negative infinity. Of X to the 15th and now all we need to think about is how to find this limit. Now what do we know? Well, recall, OK, that for a positive integer N. OK, for a positive integer and. And for a polynomial P X, OK, that's equal to AN multiplied by X N plus AN minus 1 multiplied by X N minus 1. And no far po polynomial goes all the way down to A2 X2 plus a 1 X plus a 0, OK, where A N is not equal to 0, then by the theorem for limits at infinity of powers and polynomials, if the limit. Of the limit as X approaches negative infinity of N of X to the N. OK, if we're finding the limit of this term, then it's going to be equal to negative infinity when n. Is odd. Now in this case we can tell here that 15 is odd, so the limit then of x 15th as X approaches negative infinity is going to be equal to negative infinity. Therefore, C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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3:01 minutes

Problem 2.22

Textbook Question

Determine the following limits.
lim x→−∞ (3x⁷ + x²)

Verified step by step guidance

Identify the dominant term in the expression as \(x\) approaches \(-\infty\). In this case, the dominant term is \(3x^7\) because it has the highest power.

Consider the behavior of the dominant term \(3x^7\) as \(x\) approaches \(-\infty\). Since the exponent is odd, \(3x^7\) will tend towards \(-\infty\).

Recognize that the term \(x^2\) becomes negligible compared to \(3x^7\) as \(x\) approaches \(-\infty\).

Conclude that the limit of the entire expression \(3x^7 + x^2\) is determined by the behavior of the dominant term \(3x^7\).

Therefore, the limit \(\lim_{x \to -\infty} (3x^7 + x^2)\) is the same as \(\lim_{x \to -\infty} 3x^7\), which tends towards \(-\infty\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how polynomial functions behave for very large or very small values of x. In this case, we analyze how the dominant term in the polynomial influences the limit as x approaches negative infinity.

Dominant Term

In polynomial functions, the dominant term is the term with the highest degree, as it has the most significant impact on the function's value for large magnitudes of x. For the limit in question, the term 3x^7 is the dominant term, and it will dictate the behavior of the function as x approaches negative infinity, overshadowing the lower degree term x^2.

Polynomial Behavior

Understanding polynomial behavior is essential for evaluating limits. Polynomials are continuous and smooth functions, and their end behavior can be predicted based on the leading term. In this case, since the leading term is of odd degree and has a negative coefficient when x approaches negative infinity, the overall limit will trend towards negative infinity.

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