# 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 β) (4π₯ + 5)/(5 β4π₯).

01:54

### Video Transcript

Find the limit as π₯ tends to β of four π₯ plus five over five minus four π₯.

Weβre asked to find the limit as π₯ tends to β of a quotient or rational function four π₯ plus five over five minus four π₯. Letβs call our function π of π₯, where the numerator, π of π₯, is four π₯ plus five and the denominator, π of π₯, is five minus four π₯. π of π₯ and π of π₯ are polynomials.

To find the limit as π₯ tends to β of π of π₯, we first compare the degree of π of π₯ to the degree of π of π₯. Remember that the degree of a polynomial in π₯ is the highest exponent of π₯. Since π of π₯ is four π₯ plus five, which is actually four π₯ to the power one plus five, the degree of π of π₯ is one. π of π₯ is equal to five minus four π₯, which is actually five minus four π₯ to the power one. So the degree of π of π₯ is also equal to one.

If the degree of π of π₯ is the degree of π of π₯, which is equal to π, then the limit as π₯ tends to π of π of π₯ over π of π₯ is π of π. Where π is the coefficient of the highest power of π₯ in π of π₯ and π is the coefficient of the highest power of π₯ in π of π₯.

What this means is that, to find the limit as π₯ tends to β of π of π₯. Since the degree of π of π₯ is the same as that for π of π₯, we simply divide the coefficients of the terms in π of π₯ and π of π₯, which have the largest exponent. The term with the largest exponent in π of π₯ is four π₯. And the term with the largest exponent in π of π₯ is negative four π₯. So that the limit as π₯ tends to β of our function is four divided by negative four. And thatβs equal to negative one. So that the limit as π₯ tends to β of four π₯ plus five over five minus four π₯ is negative one.