Question

Suppose \(\lim _{x \rightarrow a} f(x)=\infty\) and \(\lim _{x \rightarrow a} g(x)=\infty .\) Prove that \(\lim _{x \rightarrow a}(f(x)+g(x))=\infty\).

Short answer

Question: Prove that if \(\lim_{x\to a} f(x) = \infty\) and \(\lim_{x\to a} g(x) = \infty\), then \(\lim_{x\to a}(f(x) + g(x)) = \infty\).

Step-by-step solution

Given function limits

We are given the following limits: 1. \(\lim_{x\to a} f(x) = \infty\) 2. \(\lim_{x\to a} g(x) = \infty\) We need to prove that: \(\lim_{x\to a}(f(x) + g(x)) = \infty\).

Set up inequality for \(f(x)\) and \(g(x)\)

Since the limit of \(f(x)\) and \(g(x)\) tend to infinity as \(x\) approaches \(a\), for any large positive real number \(M_1\) and \(M_2\), there exist \(\delta_1\) and \(\delta_2 > 0\) respectively such that if \(0 < |x - a| < \delta_1\), then \(f(x) > M_1\) and if \(0 < |x - a| < \delta_2\), then \(g(x) > M_2\).

Set up inequality for \(f(x)+g(x)\)

Now, note that if \(0 < |x-a| < \delta_1,\) then \(f(x) > M_1\), and if \(0 < |x-a| < \delta_2,\) then \(g(x) > M_2\). Then: $$0 < |x-a| < \min{\{\delta_1, \delta_2}\} \Rightarrow f(x) + g(x) > M_1 + M_2$$

Prove the sum approaches infinity

Since we are given that \(\lim_{x\to a} f(x) = \infty\) and \(\lim_{x\to a} g(x) = \infty\), we have shown that for any large positive real number \(M_3 = M_1 + M_2\), we can find a \(\delta_3 = \min{\{\delta_1, \delta_2\}}\) such that if \(0 < |x - a| < \delta_3\), then \(f(x) + g(x) > M_3\). This is the definition of the limit \(\lim_{x\to a}(f(x) + g(x)) = \infty\). So we have proved that \(\lim_{x\to a}(f(x) + g(x)) = \infty\).

Key concepts

Infinity Limits

Infinity limits occur when a function grows without bounds as the input approaches a particular value. In our exercise, we focus on two functions, \( f(x) \) and \( g(x) \), both having infinity limits when \( x \) tends to \( a \). This means:

• The values of \( f(x) \) and \( g(x) \) can become arbitrarily large as \( x \) gets closer to \( a \), but never actually equals \( a \).
• The concept of infinity here denotes that there isn’t an upper bound to the values that \( f(x) \) and \( g(x) \) can take near \( a \).

By understanding this idea, proving that \( \lim_{x \to a}(f(x) + g(x)) = \infty \) becomes easier. As both functions independently grow infinitely large, their sum naturally also grows infinitely. This is a core principle when working with limits that denote infinity.

Function Limits

Function limits are all about understanding how a function behaves as its input approaches a specific point, which might not necessarily be within the function's domain. When we consider limits, we often explore:

• The output values of the function as the input nears a particular number, from the left, the right, or both sides.
• The continuous behavior of the function or any potential discontinuities that define its limits.

In the context of our exercise, the function limits of \( f(x) \) and \( g(x) \) both tend toward infinity as \( x \) approaches \( a \). For each function, even though \( x = a \) might not itself be a part of the function's range, the understanding of its limit helps in describing their behavior near \( a \). Understanding function limits is crucial in calculus as it lays the groundwork for understanding derivatives and integral calculus, highlighting points of discontinuity or rapid growth.

Epsilon-Delta Definition

The epsilon-delta definition is core to the formal understanding of limits in calculus, providing a rigorous mathematical approach. It ensures that we can make any function close enough to a limit over a certain interval:

• "Epsilon" (\( \varepsilon \)) represents how close the function value should be to the limit.
• "Delta" (\( \delta \)) indicates the distance from \( a \), the point of approach, where this closeness needs to be maintained.

In proving our exercise, we take fixed large values for the functions (\( M_1 \), \( M_2 \)), and find corresponding intervals (\( \delta_1 \), \( \delta_2 \)) where \( f(x) > M_1 \) and \( g(x) > M_2 \). We use the minimum of these intervals to ensure both functions exceed our chosen large values simultaneously, proving the sum goes to infinity. This approach outlines the infinitesimal adjustments made between the distances \( x \) can approach \( a \) (\( \delta \)) and the closeness \( f(x) \) and \( g(x) \) can achieve to infinity (since they grow larger than any selected values), demonstrating the connection and rigor the epsilon-delta definition brings to calculus.