Math Problem Statement

Compute the following limit using L'Hôpital's rule if appropriate. \( \lim_{x \to 0^+} 3 \sin(x) \ln(x) \)

Solution

To solve this limit:

limx0+3sin(x)ln(x),\lim_{x \to 0^+} 3 \sin(x) \ln(x),

we need to examine the behavior of each component as x0+x \to 0^+.

Step-by-Step Solution

  1. Analyze the components of the expression:

    • As x0+x \to 0^+, sin(x)x\sin(x) \approx x, because sin(x)\sin(x) is close to xx for small values of xx.
    • Therefore, 3sin(x)ln(x)3xln(x)3 \sin(x) \ln(x) \approx 3 x \ln(x) as x0+x \to 0^+.
  2. Rewrite the expression:

    • The limit can now be expressed as: limx0+3xln(x).\lim_{x \to 0^+} 3 x \ln(x).
  3. Identify the indeterminate form:

    • As x0+x \to 0^+, ln(x)\ln(x) \to -\infty and x0+x \to 0^+. Thus, xln(x)x \ln(x) is of the indeterminate form 0()0 \cdot (-\infty).
  4. Rewrite using L'Hôpital's Rule:

    • To apply L'Hôpital's Rule, rewrite xln(x)x \ln(x) as ln(x)1x\frac{\ln(x)}{\frac{1}{x}}.
    • Then, the limit becomes: limx0+3ln(x)1x.\lim_{x \to 0^+} 3 \frac{\ln(x)}{\frac{1}{x}}.
    • This is now in the form \frac{-\infty}{\infty}, which is suitable for L'Hôpital's Rule.
  5. Apply L'Hôpital's Rule:

    • Differentiate the numerator and the denominator with respect to xx:
      • The derivative of ln(x)\ln(x) is 1x\frac{1}{x}.
      • The derivative of 1x\frac{1}{x} is 1x2-\frac{1}{x^2}.
    • Thus, we have: limx0+31x1x2=limx0+3x=limx0+3x.\lim_{x \to 0^+} 3 \frac{\frac{1}{x}}{-\frac{1}{x^2}} = \lim_{x \to 0^+} 3 \cdot -x = \lim_{x \to 0^+} -3x.
  6. Evaluate the final limit:

    • As x0+x \to 0^+, 3x0-3x \to 0.
    • Therefore, limx0+3sin(x)ln(x)=0.\lim_{x \to 0^+} 3 \sin(x) \ln(x) = 0.

Final Answer

limx0+3sin(x)ln(x)=0.\lim_{x \to 0^+} 3 \sin(x) \ln(x) = 0.

Would you like more details on any step, or do you have additional questions?


Here are five related questions to expand your understanding:

  1. What is the purpose of using L'Hôpital's Rule in solving limits?
  2. Why is sin(x)x\sin(x) \approx x a valid approximation as x0x \to 0?
  3. How can we recognize an indeterminate form in limits?
  4. What other methods could we use to evaluate limits besides L'Hôpital's Rule?
  5. How does L'Hôpital's Rule apply to cases involving infinity?

Tip: When applying L'Hôpital's Rule, make sure to check that the limit is indeed an indeterminate form (such as 00\frac{0}{0} or \frac{\infty}{\infty}) before proceeding with differentiation.

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Math Problem Analysis

Mathematical Concepts

Limits
L'Hôpital's Rule
Trigonometric Functions
Logarithmic Functions

Formulas

L'Hôpital's Rule: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \)
Approximation of \( \sin(x) \approx x \) as \( x \to 0 \)

Theorems

L'Hôpital's Rule

Suitable Grade Level

College Level Calculus