Evaluate the following limit:
$
\lim_{x \to 0^+} \frac{\tan\left(5x^{\frac{1}{3}}\right) \log\left(1 + 3x^2\right)}{\left(\tan^{-1}\left(3\sqrt{x}\right)\right)^2 \left(e^{5x^{\frac{4}{3}}} - 1\right)}
$
Show Hint
- \( \frac{1}{15} \)
- \( 1 \)
- \( \frac{1}{3} \)
- \( \frac{5}{3} \)
The Correct Option is C
Approach Solution - 1
Step 1: Approximate the functions for small values of \( x \).
We apply small angle approximations: \( \tan(x) \approx x \) as \( x \to 0 \), \( \tan^{-1}(x) \approx x \) as \( x \to 0 \), \( \log(1 + 3x^2) \approx 3x^2 \), \( e^x - 1 \approx x \) for small \( x \).
Substituting in the expression:
\( \tan\left(5x^{\frac{1}{3}}\right) \approx 5x^{\frac{1}{3}} \), \( \log\left(1 + 3x^2\right) \approx 3x^2 \), \( \tan^{-1}\left(3\sqrt{x}\right) \approx 3\sqrt{x} \), \( e^{5x^{\frac{4}{3}}} - 1 \approx 5x^{\frac{4}{3}} \).
Substitute these approximations into the given expression: \[ \frac{5x^{\frac{1}{3}} \cdot 3x^2}{(3\sqrt{x})^2 \cdot 5x^{\frac{4}{3}}} \]
Step 2: Simplifying the expression.
Simplify the numerator and denominator: \[ \frac{15x^{\frac{7}{3}}}{9x \cdot 5x^{\frac{4}{3}}} \] This simplifies to: \[ \frac{15x^{\frac{7}{3}}}{45x^{\frac{7}{3}}} \]
Step 3: Taking the limit as \( x \to 0 \)
. As \( x \to 0 \), the expression simplifies to: \[ \frac{15}{45} = \frac{1}{3} \]
Thus, the final answer is: \[ \frac{1}{3} \]
Approach Solution -2
Step 1:
Consider each term inside the limit separately and simplify using standard limit results.
We know that for small values of x, tan(x) ≈ x, hence the first term (tan(5x1/3)/5x1/3) → 1 as x → 0.
Step 2:
Similarly, the second term ((3√x)² / (tan⁻¹(3√x))²) → 1 because for small x, tan⁻¹(3√x) ≈ 3√x.
Step 3:
Now, (ℓ(1 + 3x²) / 3x²) → 1 as x → 0 because ℓ(1 + 3x²) ≈ 3x².
Step 4:
Also, (5x4/3 / (e5x4/3/3 - 1)) → 3 as x → 0, since the denominator approximates to 5x4/3/3 using the expansion ey - 1 ≈ y.
Step 5:
The last exponential factor (5x1/3·3x² / 5x4/3·9x) simplifies to 1/3.
Step 6:
Multiplying all the limits together, we get:1 × 1 × 1 × 3 × (1/3) = 1/3.
Final Answer:1/3
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Calculus Questions
- Let \( f(x) = 2^x - x^2, \, x \in \mathbb{R} \). If \( m \) and \( n \) are respectively the number of points at which the curves \( y = f(x) \) and \( y = f'(x) \) intersect the x-axis, then the value of \( m + n \) is
- If\[\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1 - \sin 2x} \, dx = \alpha + \beta \sqrt{2} + \gamma \sqrt{3},\]where \( \alpha \), \( \beta \), and \( \gamma \) are rational numbers, then \( 3\alpha + 4\beta - \gamma \) is equal to ______.
- Let \( f \) be a differentiable function in the interval \( (0, \infty) \) such that \( f(1) = 1 \) and \(\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1\) for each \( x > 0 \).
Then \( 2f(2) + 3f(3) \) is equal to _________. The area of the region enclosed by the parabolas \( y = x^2 - 5x \) and \( y = 7x - x^2 \) is _________.
- For the function $f(x) = (\cos x) - x + 1, x \in \mathbb{R}$, find the correct relationship between the following two statements
(S1) $f(x) = 0$ for only one value of x is $[0, \pi]$.
(S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,