Find the limit of -2x - 4 as x approaches negative infinity.

Step-by-step solution

1. 1

   Separate the part that can grow from the part that cannot. In $-2x-4$ only the term $-2x$ depends on $x$; the $-4$ is a fixed number. So the behaviour of the whole expression at $-\infty$ is decided entirely by $-2x$.

2. 2

   Track the sign carefully. Saying $x\to-\infty$ means $x$ takes values that are negative with ever-larger magnitude. Multiplying a large negative number by the negative coefficient $-2$ produces a large positive number, so

   $-2x\to+\infty$

   This sign flip is the whole content of the problem: the natural guess "$x\to-\infty$ so the function does too" is wrong here.

3. 3

   Add the constant back. Subtracting the fixed amount $4$ from a quantity that is growing beyond every bound does not stop it growing:

   $-2x-4\to+\infty-4=+\infty$

4. 4

   Confirm numerically. At $x=-100$: $-2(-100)-4=196$. At $x=-10000$: $19996$. At $x=-10^6$: $1999996$. The values increase without any ceiling, exactly as predicted.

5. 5

   State the answer and the general rule. The limit is $+\infty$ — the function diverges, so there is no finite limiting value. In general, for $ax+b$ the limit as $x\to-\infty$ is $+\infty$ when $a<0$ and $-\infty$ when $a>0$; here $a=-2<0$.