Question:
$ \lim_{π₯β\infty}(1+\frac{1}{x})$ is equal to
$ \lim_{π₯β\infty}(1+\frac{1}{x})$ is equal to
Updated On: Feb 10, 2025
- e
- 1/e
- 1
- $ \infty $
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Limit Definition of the Mathematical Constant e
This limit represents the definition of the mathematical constant e, which is the base of natural logarithms.
It describes the limit:
limx β β (1 + 1/x)x
As x approaches infinity, this expression is known to converge to e, a fundamental constant in mathematics.
Was this answer helpful?
0
0
Top Questions on Limits
- If \[ \lim_{x \to \infty} \left( \frac{e}{1 - e} \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha, \] then the value of \[ \frac{\log_e \alpha}{1 + \log_e \alpha} \] equals:
- Let \[ f(x) = \lim_{n \to \infty} \sum_{r=0}^{n} \left( \frac{\tan \left( \frac{x}{2^{r+1}} \right) + \tan^3 \left( \frac{x}{2^{r+1}} \right)}{1 - \tan^2 \left( \frac{x}{2^{r+1}} \right)} \right) \] Then, \( \lim_{x \to 0} \frac{e^x - e^{f(x)}}{x - f(x)} \) is equal to:
- If \[ \lim_{x \to \infty} \left( \frac{e}{1 - e} \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha, \] then the value of \[ \frac{\log_e \alpha}{1 + \log_e \alpha} \] equals:
- Evaluate \( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \):
- $\lim_{n \to \infty} \left(\frac{n}{n^2 + 1^2} + \frac{n}{n^2 + 2^2} + \dots + \frac{n}{n^2 + 3^2 + \dots + \frac{1}{5n}} \right) =$
View More Questions
Questions Asked in IIT JAM EN exam
- Suppose an Indian company borrowed 300 dollars from a foreign bank at the beginning of the year and repaid it in dollars along with the agreed interest rate of 12 percent per annum. At the time of borrowing, the exchange rate was Rs. 70 per dollar. Assuming zero inflation rate in both the countries, the real cost of borrowing will be zero if the exchange rate is Rs. per dollar at the time of repayment (rounded off to one decimal place).
- IIT JAM EN - 2024
- External Sector and Currency Exchange rate
- The supply curve is given as $π = 10 + π₯ + 0.1π₯^2$, where π is the market price and π₯ is the quantity of goods supplied. The change in the producer surplus due to an increase in market price from 30 to 70 is ____. (rounded off to nearest integer).
- IIT JAM EN - 2024
- Supply Curve
- There are two goods π and π and there are two consumers π΄ and π΅ in a pure exchange economy. π΄ and π΅ have Cobb-Douglas utility functions of the form $π_π΄ = 2π^{0.4} π^{0.6}$ and $π_π΅ = π^{0.3}π^{0.7}$, respectively. Initially, π΄ is endowed with 50 units of good π and 20 units of good π. Similarly, π΅ is endowed with 50 units of good π and 20 units of good π. If the unit price of good π is normalised to 1, then the equilibrium unit price for good π is _____. (rounded off to two decimal places).
- IIT JAM EN - 2024
- Public goods and market failure
- Consider a closed economy IS-LM model. The goods and the money market equations are respectively given as follows:
$ π = 90 + 0.8π_π β 100π + πΊ$
$π_π = 750 + 0.2π β 260π$
where, π = national income; $π_π$ = disposable income; π = total tax given by π = 5 + 0.2π; π = interest rate; πΊ = government expenditure = 300; $ π_π $ = constant money supply = 950.
The value of π at equilibrium π is _______. (rounded off to the nearest integer).- IIT JAM EN - 2024
- Money Market
- An economy produces a consumption good and also has a research sector which produces new ideas. Time is discrete and indexed by π‘ = 0, 1, 2, β¦
The production function for the consumption good is given by
$π_π‘ = π΄_π‘ πΏ_{π¦π‘}$
where, at time π‘, $πΏ_{π¦π‘}$ is the amount of consumption good produced, π΄π‘ is the stock of existing knowledge, and $πΏ_{π¦π‘}$ is the amount of labour devoted to production of consumption good. It is known that $π΄_0 = 1$.
The production function for new ideas is given by
$ π΄_{π‘+1} β π΄_π‘ = \frac{1}{250} π΄_π‘ $πΏ_{ππ‘}
where πΏππ‘ is the amount of labour devoted to production of new ideas at time π‘. Suppose that for all π‘, $πΏ_{aπ‘}$ = 10 and $πΏ_{π¦π‘}$ = 90. Then, the growth rate of the consumption good ($Y_{π‘}$) at π‘ = 50 is _____ percent (in integer).- IIT JAM EN - 2024
- Consumption Goods
View More Questions