Determine the following limits. lim x→∞ (3x 12 − 9x 7 )

Welcome back, everyone. In this problem, we want to find the limit as x approaches infinity of 15 X 11th minus 30 X 9th. A says it's infinity, B negative infinity, C 0, and the D says it's negative 15. Now what do we know about our polynomial function here that can help us to figure out the limit at infinity? Well, the limit of the polynomial function is determined by the behavior of the leading term. In this case, it is 15 x 11th, so the limit as X approaches infinity of this expression is really going to be equal to the limit as x approaches infinity of 15 x 11th. We can break out our constant here to leave this as 15 multiplied by the limit as x approaches infinity of X to the 11th. So now, to solve, we just need to figure out what that limit is for X 11th. How can we solve? Well, recall, OK, that for a positive integer N. OK, a positive integer N. And a polynomial, OK. And a polynomial P of X, OK. Where PF X is in the form A and X to the N plus A and minus 1. Multiplied by x to the n minus 1 plus all the way down to a 2x2 plus a 1 X plus a 0. OK, where a n is not equal to 0. Then according to the theorem for limits at infinity of powers and polynomials, then the limit as X approaches infinity. Of X to the N equals infinity, OK, when N is odd. Now this applies to our problem here because for x 11thn would be equal to 11 thus n is odd. And since 11 is odd, that must mean then that this limit is going to be equal to infinity. Therefore, our limit is infinity. A 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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2:42 minutes

Problem 2.21

Textbook Question

Determine the following limits.
lim x→∞ (3x¹² − 9x⁷)

Verified step by step guidance

Identify the highest power of x in the expression, which is x^{12}.

Factor out x^{12} from the expression: 3x^{12} - 9x^{7} = x^{12}(3 - 9x^{-5}).

Rewrite the limit expression: \( \lim_{x \to \infty} x^{12}(3 - 9x^{-5}) \).

Evaluate the limit of the factored expression: \( \lim_{x \to \infty} x^{12} \) and \( \lim_{x \to \infty} (3 - 9x^{-5}) \).

Since \( x^{12} \to \infty \) and \( 3 - 9x^{-5} \to 3 \), the overall limit is determined by the behavior of x^{12}.

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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 infinity. This concept is crucial for understanding how functions behave for very large values of x, particularly in polynomial functions where the highest degree term often dominates the behavior of the function.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the context of limits, the degree of the polynomial plays a significant role in determining the limit as x approaches infinity, as the term with the highest degree will have the greatest impact on the function's value.

Dominant Term

The dominant term in a polynomial is the term with the highest degree, which significantly influences the value of the polynomial as x becomes very large. When calculating limits at infinity, identifying the dominant term allows for simplification, as lower degree terms become negligible compared to the dominant term.

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