2.1.207 AP calculus practice exam problem Lim with tan(4X)/6x

• MHB
• karush
In summary: \lim_{x\to0}\frac{4\cos4x}{4\sin4x}=\frac23\lim_{x\to0}\cot4x=\frac23\cdot\frac{1}{\tan4x}=\frac23\cdot\frac{\cos4x}{\sin4x}=\frac23\cdot\frac{1}{\frac{\sin4x}{\cos4x}}=\frac23\cdot\frac{1}{\tan4x}=\frac23\cdot\frac{1}{\frac{4x}{4x}}=\frac23\cdot\frac{1}{\frac{4x}{4x}}=\frac23\cdot\
karush
Gold Member
MHB
$\displaystyle\lim_{{x}\to{0}}\left(\frac{\tan 4x}{6x}\right)=$

(A) $\dfrac{1}{3}$

(B) $\dfrac{2}{3}$

(C) 0

(D) $-\dfrac{2}{3}$

(E) DNE

solution
[sp]
direct substitution of 0 results in undeterminant so use LH'R
so then after taking d/dx of numerator and denominator and factor out constant we have

$\displaystyle\lim_{{x}\to{0}}\left(\frac{4\sec ^2\left(4x\right)}{6}\right) =\dfrac{4}{6} \lim_{{x}\to{0}} \sec ^2(4x)$
take the limit then simplify then

$\dfrac{4}{6}\cdot 1=\dfrac{2}{3}\quad (B)$[/sp]

ok hopefully correct probable need some more itermeddiate steps

Last edited:
karush said:
$\displaystyle\lim_{{x}\to{0}}\left(\frac{\tan 4x}{6x}\right)=$

(A) $\dfrac{1}{3}$

(B) $\dfrac{2}{3}$

(C) 0

(D) $-\dfrac{2}{3}$

(E) DNE

solution
[sp]
direct substitution of 0 results in undeterminant so use LH'R
so then after taking d/dx of numerator and denominator and factor out constant we have

$\displaystyle\lim_{{x}\to{0}}\left(\frac{4\sec ^2\left(4x\right)}{6}\right) =\dfrac{4}{6} \lim_{{x}\to{0}} \sec ^2(4x)$
take the limit then simplify then

$\dfrac{4}{6}\cdot 1=\dfrac{2}{3}\quad (B)$[/sp]

ok hopefully correct probable need some more itermeddiate steps
Looks good to me. (Rock)

$$\displaystyle \frac{4x}{4x}\cdot\frac{\tan4x}{6x}=\frac{4x}{4x}\cdot\frac{\sin4x}{6x\cdot\cos{4x}}=\frac{4x}{6x}\cdot\frac{\sin4x}{\cos4x\cdot4x}$$

$$\displaystyle \frac23\lim_{x\to0}\frac{\sin4x}{\cos4x\cdot4x}=\frac23$$

1. What is the problem asking me to solve?

The problem is asking you to find the limit of the given function, tan(4X)/6x, as x approaches 0.

2. What is the purpose of this practice exam problem?

The purpose of this practice exam problem is to test your understanding of limits and your ability to solve them using calculus techniques.

3. How can I approach solving this problem?

You can approach solving this problem by using the limit definition and applying algebraic manipulation and trigonometric identities to simplify the given function.

4. What is the significance of the given function, tan(4X)/6x?

The given function, tan(4X)/6x, is significant because it involves both a trigonometric function and a polynomial function, which requires the use of different calculus techniques to solve for the limit.

5. How can I check if my answer is correct?

You can check if your answer is correct by plugging in a few values for x and evaluating the function to see if it matches your calculated limit. You can also use online calculators or ask your instructor for confirmation.

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