Math Problem Statement

Evaluate the following limit. Use​ l'Hôpital's Rule when it is convenient and applicable. ModifyingBelow lim With x right arrow 1limx→1 StartFraction ln x Over 14 x minus x squared minus 13 EndFraction lnx 14x−x2−13 Question content area bottom Part 1 Use​ l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. ModifyingBelow lim With x right arrow 1limx→1 StartFraction ln x Over 14 x minus x squared minus 13 EndFraction lnx 14x−x2−13equals=ModifyingBelow lim With x right arrow 1limx→1left parenthesis nothing right parenthesisenter your response here

Solution

To evaluate the limit

limx1lnx14xx213,\lim_{x \to 1} \frac{\ln x}{14x - x^2 - 13},

we first substitute x=1x = 1:

ln(1)=0,\ln(1) = 0, 14(1)(1)213=14113=0.14(1) - (1)^2 - 13 = 14 - 1 - 13 = 0.

This gives us the indeterminate form 00\frac{0}{0}. Thus, we can apply l'Hôpital's Rule.

Step 1: Differentiate the numerator and the denominator

  1. Numerator: The derivative of lnx\ln x is 1x\frac{1}{x}.
  2. Denominator: The derivative of 14xx21314x - x^2 - 13 is 142x14 - 2x.

Step 2: Rewrite the limit using l'Hôpital's Rule

Using l'Hôpital's Rule, we rewrite the limit:

limx1lnx14xx213=limx11x142x.\lim_{x \to 1} \frac{\ln x}{14x - x^2 - 13} = \lim_{x \to 1} \frac{\frac{1}{x}}{14 - 2x}.

Step 3: Evaluate the new limit

Now we substitute x=1x = 1:

11142(1)=1142=112.\frac{\frac{1}{1}}{14 - 2(1)} = \frac{1}{14 - 2} = \frac{1}{12}.

Conclusion

Thus, the limit evaluates to

limx1lnx14xx213=112.\lim_{x \to 1} \frac{\ln x}{14x - x^2 - 13} = \frac{1}{12}.

If you have any questions or need further details, feel free to ask!

Related Questions:

  1. How do you apply l'Hôpital's Rule in different types of indeterminate forms?
  2. What are the conditions under which l'Hôpital's Rule can be used?
  3. Can you provide examples of limits that do not require l'Hôpital's Rule?
  4. What is the significance of continuity in evaluating limits?
  5. How does the derivative of a function relate to the concept of limits?

Tip:

Always check if you can simplify the expression before applying l'Hôpital's Rule, as it may lead to an easier evaluation.

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Indeterminate Forms

Formulas

l'Hôpital's Rule
Limit definition

Theorems

l'Hôpital's Rule

Suitable Grade Level

Grades 11-12