Question:
The right hand and left hand limit of the function are respectively
The right hand and left hand limit of the function are respectively
Updated On: Jul 24, 2024
- 1 and 1
- 1 and -1
- -1 and -1
- -1 and 1
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
$LHL =\displaystyle\lim _{x \rightarrow 0^{-}}\left(\frac{e^{1 / x}-1}{e^{1 / x}+1}\right)$
$=\frac{e^{-\infty}-1}{e^{-\infty}+1}=\frac{0-1}{0+1}=-1$
Dividing both numerator and denominator by $e^{1 / x}$, we get
$\displaystyle\lim _{x \rightarrow 0}\left(\frac{1-e^{-1 / x}}{1+e^{-1 / x}}\right)$
$RHL =\displaystyle\lim _{x \rightarrow 0^{+}}\left(\frac{1-e^{-1 / x}}{1+e^{-1 / x}}\right)$
$=\frac{1-e^{-\infty}}{1+e^{-\infty}}=\frac{1-0}{1+0}=1$
$=\frac{e^{-\infty}-1}{e^{-\infty}+1}=\frac{0-1}{0+1}=-1$
Dividing both numerator and denominator by $e^{1 / x}$, we get
$\displaystyle\lim _{x \rightarrow 0}\left(\frac{1-e^{-1 / x}}{1+e^{-1 / x}}\right)$
$RHL =\displaystyle\lim _{x \rightarrow 0^{+}}\left(\frac{1-e^{-1 / x}}{1+e^{-1 / x}}\right)$
$=\frac{1-e^{-\infty}}{1+e^{-\infty}}=\frac{1-0}{1+0}=1$
Was this answer helpful?
2
0
Top Questions on Limits
- If \( f(x) = \begin{cases} \frac{(e^x - 1) \log(1 + x)}{x^2} & \text{if } x>0 \\ 1 & \text{if } x = 0 \\ \frac{\cos 4x - \cos bx}{\tan^2 x} & \text{if } x<0 \end{cases} \) is continuous at \( x = 0 \), then \(\sqrt{b^2 - a^2} =\)
- \(\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} =\)
- Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)
- Evaluate the limit: \[ \lim_{x \to 0} \frac{(\csc x - \cot x)(e^x - e^{-x})}{\sqrt{3} - \sqrt{2 + \cos x}} \]
- If a real valued function \( f(x) = \begin{cases} (1 + \sin x)^{\csc x} & , -\frac{\pi}{2} < x < 0 \\ a & , x = 0 \\ \frac{e^{2/x} + e^{3/x}}{ae^{2/x} + be^{3/x}} & , 0 < x < \frac{\pi}{2} \end{cases} \) is continuous at \(x = 0\), then \(ab = \)
View More Questions
Questions Asked in KCET exam
- The derivative of \( \sin x \) with respect to \( \log x \) is:
- KCET - 2025
- Derivatives
- If the number of terms in the binomial expansion of \((2x + 3)^n\) is 22, then the value of \(n\) is:
- KCET - 2025
- Binomial theorem
- A horizontal pipe carries water in a streamlined flow. At a point along the pipe, where the cross-sectional area is \( 10 \, \text{cm}^2 \), the velocity of water is 1 m/s and the pressure is 2000 Pa. What is the pressure of water at another point where the cross-sectional area is \( 5 \, \text{cm}^2 \)? [Density of water = 1000 kg/m³]
- KCET - 2025
- Heat Transfer
- A series LCR circuit containing an AC source of 100V has an inductor and a capacitor of reactances 24\(\Omega\) and 16\(\Omega\) respectively. If a resistance of 6\(\Omega\) is connected in series, then the potential difference across the series combination of inductor and capacitor will be
- KCET - 2025
- Electromagnetic waves
- The correct statement/s about Galvanic cell is/are:
(a) Current flows from cathode to anode
(b) Anode is positive terminal
(c) If \(E_{\text{cell}} < 0\), then it is spontaneous reaction
(d) Cathode is positive terminal- KCET - 2025
- Electrochemistry
View More Questions
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).