Is the Limit of a Polynomial as x Approaches Infinity Always Equal to Infinity? Explained with Examples

True or FalseFor any polynomial p(x), Lim p(x) = ∞x→∞

False

False. The statement is not true. The limit of a polynomial as x approaches infinity is not always equal to infinity.

To demonstrate this, let’s consider a polynomial function f(x) = 2x^2 + 3x – 1. As x approaches infinity, the behavior of f(x) can be determined by looking at the highest exponent of x in the polynomial. In this case, the highest exponent is 2.

When x approaches infinity, the 2x^2 term becomes dominant, and the other terms (3x and -1) become negligible in comparison. Therefore, we can simplify the polynomial to f(x) ≈ 2x^2 as x approaches infinity.

As x gets larger and larger, the value of f(x) grows without bound, but it does not approach positive infinity specifically. Instead, it becomes infinitely large in a positive manner. In other words, we can say that as x approaches infinity, the limit of f(x) is positive infinity, which can be written as:

Lim f(x) = ∞
x→∞

It is important to note that the limit of a polynomial can still exist even if it approaches negative infinity or oscillates between positive and negative values as x approaches infinity. Therefore, it is not accurate to always say the limit of a polynomial is equal to positive infinity.

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