Problem 44 - Calculus Early Transcendentals

Question

Suppose that \(\lim _{x \rightarrow a} f(x)=\infty\) and \(\lim _{x \rightarrow a} g(x)=c,\) where \(c\) is a real number. Prove each statement. (a) $$\lim _{x \rightarrow a}[f(x)+g(x)]=\infty$$ (b) $$\lim _{x \rightarrow a}[f(x) g(x)]=\infty \quad\( if \)c>0$$ (c) $$\lim _{x \rightarrow a}[f(x) g(x)]=-\infty\( if \)c<0$$

Step-by-Step Solution

Answer

(a) Limit is \(\infty\), (b) Limit is \(\infty\) if \(c > 0\), (c) Limit is \(-\infty\) if \(c < 0\).

Step 1: Understand the Limit of Sum

To prove that \(\lim _{x \rightarrow a}[f(x) + g(x)] = \infty\), we first note that \(\lim _{x \rightarrow a} f(x) = \infty\) implies for any large positive number \(M\), there exists \(\delta_1 > 0\) such that \(f(x) > M - c\) whenever \(0 < |x - a| < \delta_1\). Since \(\lim _{x \rightarrow a} g(x) = c\), we can also find \(\delta_2 > 0\) such that \(|g(x) - c| < \varepsilon\) for any \(\varepsilon > 0\), and we set \(\varepsilon = 0.5\), making \(g(x) > c - 0.5\). We choose \(\delta = \min(\delta_1, \delta_2)\). Then, \(f(x) + g(x) > M - c + c - 0.5 = M - 0.5\), which can be made greater than any arbitrarily large \(M\) for \(x\) near \(a\). Thus, the limit is \(\infty\).

Step 2: Understand the Limit of Product (c > 0)

To prove \(\lim _{x \rightarrow a}[f(x) g(x)] = \infty\) given \(c > 0\), observe that because \(g(x) > \frac{c}{2}\) for \(x\) near \(a\), we can write \(f(x) g(x) > f(x) \frac{c}{2}\). Since \(f(x)\) approaches infinity, \(f(x) \frac{c}{2}\) will also approach infinity for sufficiently large \(f(x)\). Thus, \(f(x) g(x)\) will approach infinity as \(x \rightarrow a\).

Step 3: Understand the Limit of Product (c < 0)

For \(\lim _{x \rightarrow a}[f(x) g(x)] = -\infty\) with \(c < 0\), observe that \(g(x) < \frac{c}{2}\) for \(x\) near \(a\). Therefore, \(f(x) g(x) < f(x) \frac{c}{2}\). As \(f(x)\) approaches infinity and \(\frac{c}{2}\) is negative, \(f(x) g(x)\) approaches \(-\infty\) as \(x \rightarrow a\), since the negative product will become infinitely large in magnitude and negative in sign.

Key Concepts

Limit of a Sum

When considering the limit of a sum, it's important to understand how each component of the sum behaves as it approaches a specific point. In our example, we're given that

• \lim _{x \rightarrow a} f(x)=\infty

which means that as \(x\) gets closer to \(a\), \(f(x)\) becomes arbitrarily large. Similarly, we know

• \lim _{x \rightarrow a} g(x)=c,

where \(c\) is some finite real number. The key idea is that the sum of an infinitely large number and a constant still tends towards infinity. This is because the infinite part \(f(x)\) dominates the expression. As a result, the later portion "> \[f(x) + g(x) > M - 0.5\] shows that no matter how large \(M\) is, we can make \(f(x) + g(x)\) exceed it as \(x\) approaches \(a\). Thus,

• \lim _{x \rightarrow a} [f(x) + g(x)] = \infty.

Limit of a Product (c > 0)

When dealing with limits involving a product, such as

• \[\lim _{x \rightarrow a} [f(x) g(x)] = \infty\] with \(c > 0\),

things are a little different than with sums. Here, the behavior of \(f(x)\) at \(x \rightarrow a\) becomes crucial since

• \(\lim _{x \rightarrow a} f(x) = \infty\)

is much larger than any positive constant \(c\). To maintain simplicity, assume \(g(x)\) is sufficiently close to \(c\). In such a scenario,

• \[f(x) g(x) > f(x) \frac{c}{2}\],

given that for \(x\) near \(a\), \(g(x)\) is slightly greater than half of \(c\). Since \(f(x)\) itself approaches infinity, the product \(f(x) \frac{c}{2}\) heads to infinity too. Therefore,

• \(\lim _{x \rightarrow a} [f(x) g(x)] = \infty\).

Limit of a Product (c < 0)

The limit of a product becomes interesting when \(c < 0\). This time, our expectation is:

• \(\lim _{x \rightarrow a} [f(x) g(x)] = -\infty\).

Observing that

• \(\lim _{x \rightarrow a} f(x) = \infty\)

means \(f(x)\) grows indefinitely. However, given

• \(g(x) < \frac{c}{2}\),
for \(x\) near \(a\),

the result changes. Since \(\frac{c}{2}\) is also negative,

• \[f(x) g(x) < f(x) \frac{c}{2}\],

indicating a product that tends toward negative infinity. This happens because we're multiplying something infinitely large with something negative. In essence,

• \(f(x)g(x)\) becomes more negatively infinite than any finite negative limit can compare, resulting in: \(\lim _{x \rightarrow a} [f(x) g(x)] = -\infty\).

Behavior Near a Point

Understanding the behavior of a function near a specific point is crucial for comprehending limits. It is essentially about examining what happens to a function \(f(x)\) as \(x\) approaches \(a\). This concept helps us determine if \(f(x)\) tends towards a real number, infinity, or negative infinity.
To visualize this, imagine walking closer to a landmark. As you approach, the landmark appears larger and larger. Similarly, as \(x\) approaches \(a\), \(f(x)\) reveals its behavior, whether it stabilizes at a number or deviates towards infinity.

• For instance, when we say

• \lim _{x \rightarrow a} f(x)=\infty,

• it means \(f(x)\) grows beyond any bounds. As such, a simple trick involves defining criteria (\(\delta\) values) that help us establish how close \(x\) needs to be to \(a\) for \(f(x)\) to behave in the expected way. By exploring these criteria, you gain a deeper understanding of how limits and proximity influence a function.