Math Problem Statement

lim((x+ln(sqrt(x^2+1)-x))/(x^3)) when x goes to 0

Solution

To solve the limit

limx0x+ln(x2+1x)x3\lim_{x \to 0} \frac{x + \ln\left(\sqrt{x^2 + 1} - x\right)}{x^3}

we can break it down and simplify it.

Step 1: Simplify the Logarithmic Term

First, consider the term inside the logarithm: x2+1x.\sqrt{x^2 + 1} - x.

Using the identity ab=ab2a+b\sqrt{a} - b = \frac{a - b^2}{\sqrt{a} + b} to simplify: x2+1x=(x2+1)x2x2+1+x=1x2+1+x.\sqrt{x^2 + 1} - x = \frac{(x^2 + 1) - x^2}{\sqrt{x^2 + 1} + x} = \frac{1}{\sqrt{x^2 + 1} + x}.

Thus, ln(x2+1x)=ln(1x2+1+x)=ln(x2+1+x).\ln\left(\sqrt{x^2 + 1} - x\right) = \ln\left(\frac{1}{\sqrt{x^2 + 1} + x}\right) = -\ln(\sqrt{x^2 + 1} + x).

Step 2: Expand ln(x2+1+x)\ln(\sqrt{x^2 + 1} + x) Around x=0x = 0

Using the Taylor series for x2+11+x22\sqrt{x^2 + 1} \approx 1 + \frac{x^2}{2} as x0x \to 0: x2+1+x1+x22+x.\sqrt{x^2 + 1} + x \approx 1 + \frac{x^2}{2} + x.

Thus, ln(x2+1+x)ln(1+x+x22).\ln(\sqrt{x^2 + 1} + x) \approx \ln\left(1 + x + \frac{x^2}{2}\right).

Using the Taylor expansion for ln(1+u)u\ln(1 + u) \approx u when u0u \to 0: ln(1+x+x22)x+x22.\ln\left(1 + x + \frac{x^2}{2}\right) \approx x + \frac{x^2}{2}.

Thus, ln(x2+1x)(x+x22).\ln\left(\sqrt{x^2 + 1} - x\right) \approx -(x + \frac{x^2}{2}).

Step 3: Substitute and Simplify the Limit Expression

Now we rewrite the original limit expression using this approximation: limx0xxx22x3=limx0x22x3.\lim_{x \to 0} \frac{x - x - \frac{x^2}{2}}{x^3} = \lim_{x \to 0} \frac{-\frac{x^2}{2}}{x^3}.

This simplifies to: limx0x22x3=limx012x=0.\lim_{x \to 0} -\frac{x^2}{2x^3} = \lim_{x \to 0} -\frac{1}{2x} = 0.

Therefore, the answer is:

0.\boxed{0}.

Would you like more details or have any questions?

Related Questions:

  1. How can we apply L'Hôpital's Rule to solve similar indeterminate forms?
  2. What are some alternative methods for simplifying limits with logarithmic terms?
  3. How does the Taylor series expansion help in evaluating limits?
  4. Can this approach be applied to higher-order terms in the Taylor series?
  5. What other indeterminate forms commonly appear in limit problems?

Tip:

When dealing with limits that involve logarithms or square roots, simplifying by rationalizing or using series expansion often makes complex expressions more manageable.

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Logarithmic functions
Taylor Series Expansion

Formulas

Limit definition for indeterminate forms
Taylor series expansion for logarithmic and square root functions around x = 0

Theorems

Taylor Series Expansion
Logarithmic Properties
Limit Laws

Suitable Grade Level

Undergraduate Calculus