Negative of Real Function that Increases Without Bound

Theorem

Let $f: \R \to \R$ be a real function.

Then:

$(1): \quad \ds \lim_{x \mathop \to +\infty} \map f x = +\infty \implies \lim_{x \mathop \to +\infty} -\map f x = -\infty$

$(2): \quad \ds \lim_{x \mathop \to -\infty} \map f x = +\infty \implies \lim_{x \mathop \to -\infty} -\map f x = -\infty$

Proof

Suppose $\ds \lim_{x \mathop \to +\infty} \map f x = +\infty$.

Then by the definition of infinite limits at infinity:

$\forall M > 0: \exists N > 0: x > N \implies \map f x > M$

But $M > 0 \iff -M < 0$.

Likewise $\map f x > M \iff -\map f x < -M$.

Putting $M' = -M$:

$\forall M' < 0: \exists N > 0: x > N \implies -\map f x < M'$

The result then follows from the definition of negative infinite limit at infinity.

The proof for $\ds \lim_{x \mathop \to -\infty} \map f x = +\infty$ is analagous.

$\blacksquare$

Also see

• Negative of Real Function that Decreases Without Bound