Let
\(β = \lim_{x →0} \frac{αx-(e^{3x}-1)}{αx(e^{3x}-1) }\)for some\( α \in R.\)
Then the value of α+β is
Let
\(β = \lim_{x →0} \frac{αx-(e^{3x}-1)}{αx(e^{3x}-1) }\)for some\( α \in R.\)
Then the value of α+β is
\(\frac{14}{5}\)
\(\frac{3}{2}\)
\(\frac{5}{2}\)
\(\frac{7}{2}\)
The Correct Option is C
Solution and Explanation
The correct answer is (C) : \(\frac{5}{2}\)
\(β = \lim _{x →0} \frac{αx-(e^{3x}-1)}{αx(e^{3x}-1)}, α \in R.\)
\(=\lim_{x →0} \frac{\frac{α}{3}-(\frac{e^{3x}-1}{3x})}{αx(\frac{e^{3x-1}}{3x})}\)
So, α = 3 (to make indeterminant form)
\(β = \lim_{x\rightarrow0}\frac{1-(\frac{e^{3x}-1}{3x})}{3x}\)
\(= \frac{1-\frac{(3x+\frac{9x^2}{2}+...)}{3x}}{3x}\)
\(= -\frac{(\frac{9}{2}x^2+\frac{(3x)^3}{31}+...)}{9x^2}=\frac{-1}{2}\)
\(∴ α+β = 3 -\frac{1}{2}\)
\(= \frac{5}{2}\)
Top Questions on Limits
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to:
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\[\begin{array}{|c|c|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline \text{(A)}\ \lim_{x\to 0}(1+2x)^{\frac{1}{x}} & \text{(I)}\ e^{6} \\ \hline \text{(B)}\ \lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x} & \text{(II)}\ e^{2} \\ \hline \text{(C)}\ \lim_{x\to 0}(1+5x)^{\frac{2}{x}} & \text{(III)}\ e \\ \hline \text{(D)}\ \lim_{x\to \infty}\left(1+\frac{3}{x}\right)^{2x} & \text{(IV)}\ e^{5} \\ \hline \end{array}\] 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:
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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).