Problem 29 - Calculus Early Transcendentals

Question

Use the precise definition of infinite limits to prove the following limits. $$\lim _{x \rightarrow 4} \frac{1}{(x-4)^{2}}=\infty$$

Step-by-Step Solution

Answer

Question: Prove that the limit of the function f(x) = 1/(x-4)^2 as x approaches 4 is infinity using the precise definition of infinite limits. Answer: The limit of the function f(x) = 1/(x-4)^2 as x approaches 4 is infinity, because for any M > 0, we can find a δ > 0 such that if 0 < |x-4| < δ, then f(x) > M. In particular, we can choose δ = 1/√M.

Step 1: Write the given function and the limit

We have the function f(x) = 1/(x-4)^2, and we need to prove that $$\lim _{x \rightarrow 4} \frac{1}{(x-4)^{2}}=\infty $$

Step 2: Apply the precise definition of infinite limits

According to the precise definition, we have f(x) > M when 0 < |x-4| < δ for some δ > 0. First, let's rewrite f(x) > M as the inequality: $$ \frac{1}{(x-4)^2} > M $$

Step 3: Manipulate the inequality to isolate δ

Now, we will manipulate the inequality to isolate δ, which involves solving it for (x-4). Start by taking the reciprocal of both sides: $$ (x-4)^2 < \frac{1}{M} $$ Since (x-4)^2 is always positive, we can take the square root of both sides: $$ |x-4| < \sqrt{\frac{1}{M}} $$

Step 4: Choose the value of δ from the inequality

From the inequality above, we can choose δ = 1/√M. For every M > 0, there always exists a δ > 0 such that when 0 < |x-4| < δ, then f(x) > M.

Step 5: Conclusion

By applying the precise definition of infinite limits and finding the appropriate value of δ for every M > 0, we have proven that: $$\lim _{x \rightarrow 4} \frac{1}{(x-4)^{2}}=\infty$$

Key Concepts

Limit Definition

Understanding the limit definition is crucial in calculus. The limit of a function describes the behavior of that function as the input gets closer to a particular point. In the exercise above, we're dealing with an infinite limit. This means as x approaches 4, the function's value, \(\frac{1}{(x-4)^2}\), increases without bound.

Here, we say the limit is infinite because for every large number \(M\), we can find an interval around x = 4 where the function exceeds \(M\). This is represented mathematically by the expression:

• \(\lim _{x \rightarrow 4} \frac{1}{(x-4)^{2}}=\infty\)

Understanding that the function never actually "reaches" infinity but grows larger and larger as x moves toward 4 is key to conceptualizing infinite limits.

Epsilon-Delta Proof

The epsilon-delta proof is a formal method used to verify limit statements. Although the example given is of infinity, the logic still applies in a slightly modified form. For infinite limits, we look for a delta (\(\delta\)) such that whenever \(0 < |x-4| < \delta\), the function value \(f(x)\) is greater than some threshold \(M\).

The process involves:

• Setting up the inequality \(\frac{1}{(x-4)^2} > M\).
• Manipulating the inequality to derive a delta condition, resulting in \((x-4)^2 < \frac{1}{M}\).
• Taking the square root to isolate \(|x-4| < \sqrt{\frac{1}{M}}\).

Finally, \(\delta\) is chosen as \(\frac{1}{\sqrt{M}}\). This precise choice of \(\delta\) ensures that for every positive \(M\), there exists a \(\delta\) such that the values of \(f(x)\) are greater than \(M\), proving the infinite limit statement.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a certain point or value, particularly when it nears infinity or negative infinity. As functions near these values, their specific behavior can often be generalized.

In the given exercise, \(\frac{1}{(x-4)^2}\) tends towards infinity as \(x\) approaches 4. Here, the vertical asymptote is located at \(x = 4\). This means:

• The closer \(x\) gets to 4, the larger \(\frac{1}{(x-4)^2\) becomes.
• The function values increase rapidly with small changes in \(x\).

Recognizing this pattern is vital because it reveals that the function's output becomes unbounded, characteristic of a vertical asymptote. The function never quite reaches infinity, similar to our intuitive idea of approaching a cliff's edge: close to the edge, but never over it.