Understanding Limits as x Approaches Infinity in Fractional Equations: Analyzing Powers in the Numerator and Denominator

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To find the limit as x approaches infinity of a fraction with powers in both the numerator and denominator, you can examine the terms with the highest powers in the numerator and denominator separately

To find the limit as x approaches infinity of a fraction with powers in both the numerator and denominator, you can examine the terms with the highest powers in the numerator and denominator separately.

Let’s say you have a fraction:

lim x->∞ (f(x) / g(x))

where f(x) and g(x) are functions of x.

If both f(x) and g(x) have the same highest power of x, then you can take the ratio of the coefficients of those terms to determine the limit. For example:

lim x->∞ (3x^2 + 2x + 1) / (2x^2 – x + 3)

In this case, the highest power of x in both the numerator and denominator is x^2. Taking the ratio of the coefficients of the x^2 term, we have:

lim x->∞ (3x^2 / 2x^2)

The x^2 terms cancel out, leaving us with:

lim x->∞ (3 / 2)

Therefore, the limit is 3/2 or 1.5 as x approaches infinity.

However, if the highest power of x in the numerator is greater than the highest power of x in the denominator, the limit will approach positive infinity or negative infinity, depending on the signs of the leading terms.

For example:

lim x->∞ (5x^3 + 2x^2 + 3) / (2x^2 + 4)

In this case, the highest power of x in the numerator is x^3, while the highest power of x in the denominator is x^2. Since the power of x in the numerator is greater, the limit will approach positive infinity as x approaches infinity.

On the other hand, if the highest power of x in the denominator is greater than the highest power of x in the numerator, the limit will approach 0.

For example:

lim x->∞ (2x^2 + 3) / (5x^3 + 2x + 1)

In this case, the highest power of x in the numerator is x^2, while the highest power of x in the denominator is x^3. Since the power of x in the denominator is greater, the limit will approach 0 as x approaches infinity.

In summary, when finding the limit as x approaches infinity of a fraction with powers in the numerator and denominator, compare the highest powers of x in each and consider the ratios or signs of the leading terms to determine if the limit approaches a specific number, positive infinity, negative infinity, or 0.

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