Limit of Rational Function...1?

f(x) = (2x + 1)/(x + 4)

We can re-write this as 2(x + 4)/(x + 4) - 7/(x + 4)

i.e. 2 - [7/(x + 4)]

As x --> ±∞, 7/(x + 4) --> 0

Hence, the limit tends to 2

As you say, x = -4 is a vertical asymptote, so we need to establish the behaviour of f(x) for values less than -4 and values more than -4

If we choose x < -4, i.e. negative values, we have:

f(-ve) = 2 - [7/(-ve)] => 2 + (+ve)

This means the values of f(x) are greater than 2

Hence, for values less than x = -4, f(x) approaches a limit of 2 from above

Choosing x > -4, we have:

2 - [7/(+ve)] => 2 - (+ve)

This means the values of f(x) are less than 2

Hence, for values greater than x = -4, f(x) approaches a limit of 2 from below.

No need to sketch the graph, but it does help us to visualise the above.

:)>

The numerator approaches 2(-4) + 1 = -8 + 1 = -7

As x approaches -4 from above ("the right side"), then x + 4 approaches 0 from above, meaning the denominator is positive.

With the numerator negative and the denominator positive, the overall result is negative.

With the denominator approaching zero, and the numerator NOT approaching zero, that means the overall value approaches positive or negative infinity.

Put those together and the answer is negative infinity.

No graphing is necessary.