Video: Finding the Limit as π‘₯ Approaches Infinity of a Rational Function Where the Numerator and Denominator Are of the Same Degree

Find lim_(π‘₯ β†’ ∞) (4π‘₯ + 5)/(5 βˆ’4π‘₯).

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Video Transcript

Find the limit as π‘₯ tends to ∞ of four π‘₯ plus five over five minus four π‘₯.

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

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

If the degree of 𝑃 of π‘₯ is the degree of 𝑄 of π‘₯, which is equal to 𝑛, then the limit as π‘₯ tends to π‘Ž of 𝑃 of π‘₯ over 𝑄 of π‘₯ is 𝑃 of 𝑄. Where 𝑃 is the coefficient of the highest power of π‘₯ in 𝑃 of π‘₯ and 𝑄 is the coefficient of the highest power of π‘₯ in 𝑄 of π‘₯.

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

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