3.4.238 AP calculus exam Limits with ln

In summary, the given limit expression is $\displaystyle\lim_{h\to 0}\dfrac{\ln{(4+h)}-\ln{h}}{h}$ and the options are (A) 0, (B) $\dfrac{1}{4}$, (C) 1, (D) $e$, and (E) DNE. After checking the limit definition of a derivative, it is determined that the limit diverges, making the answer (E) DNE.
  • #1
karush
Gold Member
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Solve
$$\displaystyle\lim_{h\to 0}
\dfrac{\ln{(4+h)}-\ln{h}}{h}$$
$$(A)\,0\quad
(B)\, \dfrac{1}{4}\quad
(C)\, 1\quad
(D)\, e\quad
(E)\, DNE$$
The Limit diverges so the Limit Does Not Exist (E)ok the only way I saw that it diverges is by plotting
not sure what the rule is that observation would make ploting not needed

also I noticed these AP sample questions are getting a lot of views so thot I would continue to post more
 
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  • #2
Check your limit expression again. Should be

$\displaystyle \lim_{h \to 0} \dfrac{\ln(4+h) - \color{red}{ \ln(4)}}{h}$
Recall the limit definition of a derivative ...

$\displaystyle f’(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$

let $x=4$

... try again.
 

FAQ: 3.4.238 AP calculus exam Limits with ln

1. What is the concept of limits in AP Calculus?

Limits in AP Calculus refer to the value that a function approaches as the input approaches a certain value. It is used to determine the behavior of a function at a specific point and is a fundamental concept in calculus.

2. How do you solve limits with logarithmic functions?

To solve limits with logarithmic functions, you can use the rule that the limit of a logarithmic function is equal to the logarithm of the limit of the function. You can also use L'Hopital's rule or algebraic manipulation to simplify the function before taking the limit.

3. Can ln be used to solve limits involving exponential functions?

Yes, ln (natural logarithm) can be used to solve limits involving exponential functions. You can use the fact that ln and e (natural base) are inverse functions to simplify the limit and solve for the value.

4. Are there any special cases when solving limits with ln?

Yes, there are a few special cases when solving limits with ln. One is when the limit approaches 0, in which case the limit will be equal to negative infinity. Another is when the limit approaches infinity, in which case the limit will be equal to infinity.

5. How is the concept of limits with ln used in real-life applications?

Limits with ln are commonly used in real-life applications to model growth and decay. For example, in finance, the natural logarithm is used to calculate compound interest. In biology, it is used to model population growth. In physics, it is used to model radioactive decay. Understanding limits with ln is essential for solving these types of real-world problems.

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