# Video: Finding the Limit as 𝑥 Approaches Infinity of a Rational Function Where the Numerator and Denominator Are of the Same Degree

Find lim_(𝑥 → ∞) (4𝑥 + 5)/(5 −4𝑥).

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### Video Transcript

Find the limit as 𝑥 tends to ∞ of four 𝑥 plus five over five minus four 𝑥.

We’re asked to find the limit as 𝑥 tends to ∞ of a quotient or rational function four 𝑥 plus five over five minus four 𝑥. Let’s call our function 𝑓 of 𝑥, where the numerator, 𝑃 of 𝑥, is four 𝑥 plus five and the denominator, 𝑄 of 𝑥, is five minus four 𝑥. 𝑃 of 𝑥 and 𝑄 of 𝑥 are polynomials.

To find the limit as 𝑥 tends to ∞ of 𝑓 of 𝑥, we first compare the degree of 𝑃 of 𝑥 to the degree of 𝑄 of 𝑥. Remember that the degree of a polynomial in 𝑥 is the highest exponent of 𝑥. Since 𝑃 of 𝑥 is four 𝑥 plus five, which is actually four 𝑥 to the power one plus five, the degree of 𝑃 of 𝑥 is one. 𝑄 of 𝑥 is equal to five minus four 𝑥, which is actually five minus four 𝑥 to the power one. So the degree of 𝑄 of 𝑥 is also equal to one.

If the degree of 𝑃 of 𝑥 is the degree of 𝑄 of 𝑥, which is equal to 𝑛, then the limit as 𝑥 tends to 𝑎 of 𝑃 of 𝑥 over 𝑄 of 𝑥 is 𝑃 of 𝑄. Where 𝑃 is the coefficient of the highest power of 𝑥 in 𝑃 of 𝑥 and 𝑄 is the coefficient of the highest power of 𝑥 in 𝑄 of 𝑥.

What this means is that, to find the limit as 𝑥 tends to ∞ of 𝑓 of 𝑥. Since the degree of 𝑃 of 𝑥 is the same as that for 𝑄 of 𝑥, we simply divide the coefficients of the terms in 𝑃 of 𝑥 and 𝑄 of 𝑥, which have the largest exponent. The term with the largest exponent in 𝑃 of 𝑥 is four 𝑥. And the term with the largest exponent in 𝑄 of 𝑥 is negative four 𝑥. So that the limit as 𝑥 tends to ∞ of our function is four divided by negative four. And that’s equal to negative one. So that the limit as 𝑥 tends to ∞ of four 𝑥 plus five over five minus four 𝑥 is negative one.