if `lim_(x->0)(ptanqx^2-3cos^2x+4)^(1/(3x^2))=e^(5/3)`

Options

A. `p=sqrt2,q= (1)/(2 sqrt2)`
B. `=(1)/(sqrt2), q=2 sqrt2`
C. `p=1,q=2`
D. `p=2,q=4`

Answer

To solve the limit problem given by:

\[ \lim_{x \to 0} \left( p \tan(q x^2) - 3 \cos^2(x) + 4 \right)^{\frac{1}{3x^2}} = e^{\frac{5}{3}} \]

we will follow these steps:

### Step 1: Analyze the limit expression
As \( x \to 0 \), we can evaluate the expression inside the limit. We know that:

- \( \tan(q x^2) \approx q x^2 \) (since \( \tan(x) \approx x \) for small \( x \))
- \( \cos^2(x) \approx 1 \) (since \( \cos(x) \approx 1 \) for small \( x \))

Substituting these approximations into the limit gives:

\[ p \tan(q x^2) - 3 \cos^2(x) + 4 \approx p(q x^2) - 3(1) + 4 = pq x^2 + 1 \]

### Step 2: Rewrite the limit
Now we can rewrite the limit as:

\[ \lim_{x \to 0} \left( pq x^2 + 1 \right)^{\frac{1}{3x^2}} \]

### Step 3: Identify the form of the limit
As \( x \to 0 \), \( pq x^2 + 1 \) approaches \( 1 \), and thus we have the indeterminate form \( 1^\infty \). We can use the property of limits:

\[ \lim_{x \to 0} f(x)^{g(x)} = e^{\lim_{x \to 0} g(x)(f(x) - 1)} \]

Here, let \( f(x) = pq x^2 + 1 \) and \( g(x) = \frac{1}{3x^2} \).

### Step 4: Calculate \( g(x)(f(x) - 1) \)
We find:

\[ f(x) - 1 = pq x^2 \]

Thus,

\[ g(x)(f(x) - 1) = \frac{1}{3x^2}(pq x^2) = \frac{pq}{3} \]

### Step 5: Evaluate the limit
Now we can evaluate the limit:

\[ \lim_{x \to 0} g(x)(f(x) - 1) = \frac{pq}{3} \]

So we have:

\[ \lim_{x \to 0} \left( pq x^2 + 1 \right)^{\frac{1}{3x^2}} = e^{\frac{pq}{3}} \]

### Step 6: Set the limit equal to the given expression
We know from the problem statement that this limit equals \( e^{\frac{5}{3}} \):

\[ e^{\frac{pq}{3}} = e^{\frac{5}{3}} \]

### Step 7: Equate the exponents
Since the bases are the same, we can equate the exponents:

\[ \frac{pq}{3} = \frac{5}{3} \]

### Step 8: Solve for \( pq \)
Multiplying both sides by 3 gives:

\[ pq = 5 \]

### Step 9: Find possible values of \( p \) and \( q \)
We need to find pairs \( (p, q) \) such that \( pq = 5 \). Possible pairs include:

- \( (1, 5) \)
- \( (5, 1) \)
- \( (2, 2.5) \)
- \( (2.5, 2) \)

### Conclusion
The values of \( p \) and \( q \) must satisfy \( pq = 5 \).

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