T$\displaystyle\lim_{x \to 0+}$ $\left(x^{n} \,ln\, x\right), n > 0$
- does not exist
- exists and is zero
- exists and is 1
- exists and is $e^{-1}$
The Correct Option is B
Solution and Explanation
$\Rightarrow\, \displaystyle\lim_{x \to 0+} \frac{\frac{1}{x}}{\frac{-n}{x^{n+1}}} = 0$
Top Questions on Limits
- Match List-I with List-II:
\[\begin{array}{|c|c|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline \text{(A)}\ \lim_{x\to 0}(1+2x)^{\frac{1}{x}} & \text{(I)}\ e^{6} \\ \hline \text{(B)}\ \lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x} & \text{(II)}\ e^{2} \\ \hline \text{(C)}\ \lim_{x\to 0}(1+5x)^{\frac{2}{x}} & \text{(III)}\ e \\ \hline \text{(D)}\ \lim_{x\to \infty}\left(1+\frac{3}{x}\right)^{2x} & \text{(IV)}\ e^{5} \\ \hline \end{array}\] Choose the correct answer from the options given below: - The value of $\displaystyle \lim_{x \to \infty}\left(1+\frac{2}{3x}\right)^{x}$ is:
- If $ \lim_{x \to 0} \frac{\cos(2x) + a \cos(4x) - b}{x^4} $ is finite, then $ (a + b) $ is equal to:
For \( \alpha, \beta, \gamma \in \mathbb{R} \), if \[ \lim_{x \to 0} \frac{x^2 \sin(\alpha x) + (\gamma - 1)e^{x^2}}{\sin(2x - \beta x)} = 3, \] then \( \beta + \gamma - \alpha \) is equal to:
- For $ t>-1 $, let $ \alpha_t $ and $ \beta_t $ be the roots of the equation $ \left( (t + 2)^{\frac{1}{7}} - 1 \right)x^2 + \left( (t + 2)^{\frac{1}{6}} - 1 \right)x + \left( (t + 2)^{\frac{1}{21}} - 1 \right) = 0. $ If $ \lim_{t \to 1^+} \alpha_t = a $ and $ \lim_{t \to 1^+} \beta_t = b $, then $ 72(a + b)^2 $ is equal to:
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).