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T$\displaystyle\lim_{x \to 0+}$ $\left(x^{n} \,ln\, x\right), n > 0$
T$\displaystyle\lim_{x \to 0+}$ $\left(x^{n} \,ln\, x\right), n > 0$
Updated On: Apr 27, 2024
- does not exist
- exists and is zero
- exists and is 1
- exists and is $e^{-1}$
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The Correct Option is B
Solution and Explanation
$\displaystyle\lim_{x \to 0+}$$\frac{\ell nx}{\frac{1}{x^{n}}}\left(\frac{\infty}{\infty}\right).$ Applying LH rule
$\Rightarrow\, \displaystyle\lim_{x \to 0+} \frac{\frac{1}{x}}{\frac{-n}{x^{n+1}}} = 0$
$\Rightarrow\, \displaystyle\lim_{x \to 0+} \frac{\frac{1}{x}}{\frac{-n}{x^{n+1}}} = 0$
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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).
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