Problem 45 - Calculus with Concepts in Calculus

Question

It follows from the Substitution Rule that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(1 / x)\) and \(\lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow 0^{-}} f(1 / x)\) Use these formulas to evaluate the limit. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}}{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Limit to Evaluate

We need to evaluate the limit \( \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}}{x} \). This involves finding what happens to the expression as \( x \) approaches infinity.

Step 2: Apply the Substitution Rule

According to the substitution rule, \( \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}}{x} = \lim _{x \rightarrow 0^{+}} \frac{\sqrt{1+(1/x)^{2}}}{1/x} \). Here, substitute \( x \) with \( 1/u \) where \( u \to 0^{+} \).

Step 3: Simplify the Expression

The expression becomes \( \lim _{u \to 0^{+}} \frac{\sqrt{1+u^{2}}}{u^{-1}} \). Simplify it to \( \lim _{u \to 0^{+}} u \sqrt{1+u^{2}} \).

Step 4: Evaluate the Limit as \( u \to 0^{+} \)

As \( u \to 0^{+} \), \( u \sqrt{1+u^{2}} \to 0 \times \sqrt{1+0^{2}} = 0 \). Therefore, \( \lim _{u \to 0^{+}} u \sqrt{1+u^{2}} = 0 \).

Key Concepts

Substitution Rule

The Substitution Rule in calculus helps simplify the process of evaluating limits, especially when looking at limits at infinity. By swapping a variable that approaches infinity with a more manageable one that approaches zero or some other value, we can transform a complex expression into a simpler form. In this exercise, the substitution rule is applied to the function \(f(x)\) such that when \(x\) approaches infinity, we can examine \(f(1/x)\) as \(x\) approaches zero:

• This is useful for making expressions easier to evaluate.
• It allows focusing on the behavior of \(f(x)\) around zero rather than infinity.
• The substitution here is \(x = 1/u\) where \(u \to 0^{+}\).

This transformation turns a potentially unwieldy limit problem into one that's more familiar and ties back to principles of continuity and behavior near zero.

Limit at Infinity

When considering limits at infinity, we're essentially asking what happens to a function as the input grows very large, in either a positive or negative direction. The goal is to determine the function's behavior as it reaches infinity. For the given function, \(\lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}}{x}\):

• We think about the dominating terms as \(x\) becomes extremely large.
• Identifying these can often simplify the function; here, \(x^2\) becomes the focus inside the square root as \(x\) increases.
• The substitution proposed earlier allows focusing on small values instead, turning the problem of infinity into one of near-zero behavior.

These insights can significantly ease the evaluation process for limits at infinity, as limiting behavior becomes transparently obvious with large \(x\).

Simplification of Expressions

Simplification is a vital step in solving limit problems, especially those involving complex algebraic forms. After substitution, our original expression \(\frac{\sqrt{1+(1/x)^{2}}}{1/x}\) needs to be simplified:

• Rewriting it as \(u \sqrt{1+u^{2}}\) highlights how terms behave as they grow small.
• Simplification ensures that each component of the expression is studied effectively, focusing on significant elements of the function.
• In this case, allowing \(u\) to approach zero simplifies to assessing \(0 \times \sqrt{1 + 0^2}\), straightforwardly revealing the limit to be zero.

Through simplification, complicated expressions can be broken down, enabling efficient evaluation of limits with clearer insight.