Question

Suppose \(\lim _{x \rightarrow a} f(x)=\infty .\) Prove that \(\lim _{x \rightarrow a}(f(x)+c)=\infty\) for any constant \(c\).

Short answer

Question: Prove that if the limit of the function f(x) as x approaches 'a' is infinity, the limit of the function (f(x) + c), where c is any constant, as x approaches 'a' is also infinity. Solution: We showed that for any arbitrarily large M, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) + c| > M. Using the definition of a limit equaling infinity and the triangle inequality, we demonstrated that the same δ works for both f(x) and (f(x) + c). Therefore, we can conclude that if the limit of the function f(x) as x approaches 'a' is infinity, the limit of the function (f(x) + c) as x approaches 'a' is also infinity.

Step-by-step solution

Recall the definition of a limit equaling infinity

The definition of a limit equaling infinity states that for any arbitrarily large M, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x)| > M. In this problem, we are given that $$ \lim_{x \to a} f(x) = \infty $$ So we know that if 0 < |x - a| < δ, then |f(x)| > M.

Add the constant 'c' to f(x)

Now we want to consider the function (f(x) + c), with c being any given constant. To show that the limit of this function is also infinity as x approaches 'a', we need to show that: $$ \lim_{x \to a}(f(x) + c) = \infty $$

Use the definition of a limit equaling infinity

Let's work with the definition of a limit equaling infinity: For any arbitrarily large M, we must find a δ > 0 such that if 0 < |x - a| < δ, then |f(x) + c| > M. We know that if 0 < |x - a| < δ, then |f(x)| > M (from the given limit that \(\lim_{x \to a} f(x) = \infty\)). Now we need to show that this same δ works for |f(x) + c| > M. Since |f(x)| > M, it is also true that |f(x)| + |c| > M + |c|. By applying the triangle inequality, we get: $$ |f(x) + c| \ge |f(x)| - |c| > M + |c| - |c| = M $$

Conclusion

Since we have found a δ that satisfies the definition of a limit equaling infinity for the function (f(x) + c), we can conclude that $$ \lim_{x \to a}(f(x) + c) = \infty $$ for any constant c.

Key concepts

Limits at Infinity

Understanding limits at infinity is fundamental to grasping the behavior of functions as they stretch towards the endless horizon of mathematical abstraction. Essentially, when stating that \( \lim_{x \to a} f(x) = \infty \), we are conveying that as \(x\) approaches some value \(a\), the function \(f(x)\) grows without bound. This is a conceptualization of infinity—rather than a destination, it's a journey with no end.

Consider a basic graph where the x-axis represents our value \(a\) and the y-axis embodies \(f(x)\). As we move along the graph towards \(a\), if \(f(x)\) keeps rising, spiralling away from any fixed ceiling, no matter how high, it indicates that \(f(x)\) tends to infinity as it gets closer to \(a\). This concept is pivotal when evaluating limits in more complex calculus problems and understanding how functions behave at the edge of the mathematical universe.

Definition of a Limit

Delving into the definition of a limit, we encounter the rigorous language of calculus that helps us make precise the idea of a function approaching a value. The formal definition stipulates that \( \lim_{x \to a} f(x) = L \) if, for any \( \epsilon > 0 \) (no matter how tiny), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - a| < \delta \) (when \(x\) is within a \(\delta\)-neighborhood of \(a\), but not at \(a\) itself), \( |f(x) - L| < \epsilon \). In simpler terms, this means we can make \(f(x)\)'s value as close to \(L\) as we desire, by ensuring that \(x\) is close enough to \(a\).

For our current exercise, replace \(L\) with the symbol for infinity (\(\infty\)), to indicate that as \(x\) comes close to \(a\), \(f(x)\) surpasses any pre-set boundary, encapsulating the notion of unlimited growth.

Triangle Inequality

The triangle inequality is a principle from geometry that can be applied in various mathematical contexts, including calculus. It asserts that for any real numbers \(a\) and \(b\), the absolute value of their sum is always less than or equal to the sum of their absolute values: \( |a + b| \le |a| + |b| \). This property helps in understanding the distance between points on a number line and is quintessential when working with complex numbers and vectors.

In the context of proving limits, the triangle inequality assists in demonstrating the behavior of \( |f(x) + c| \), where \(c\) is a constant, as part of the step by step analysis. It allows us to firmly state that if \( |f(x)| > M \) for some M, then \( |f(x) + c| \) will also exceed M (since \( M + |c| - |c| \) equals M, a helpful manipulation using the triangle inequality to simplify the expression).

Proving Limits

The process of proving limits involves verifying that a function conforms to the precise definition of a limit. In practice, this often requires a creative blend of algebraic manipulation and logical reasoning, invoking the limit definition and the properties of inequalities, as in the triangle inequality. Proving that a limit equals infinity, as in our exercise, means showing that for any large arbitrary number (\(M\)), we can locate a window (\(\delta\)-neighborhood) around \(a\) such that \(f(x)\) exceeds \(M\) within this window.

Applied to our exercise, we began with the proposition that \( \lim_{x \to a} f(x) = \infty \). We then reasoned that adding a constant \(c\) would not change the fundamental behavior of the function stretching towards infinity. By carefully applying the limit definition and triangle inequality, we successfully showed that \( \lim_{x \to a} (f(x) + c) = \infty \), thereby proving the limit for any constant value added to our function.