Articles of indeterminate forms

Why don't I get $e$ when I solve $\lim_{n\to \infty}(1 + \frac{1}{n})^n$?

If I were given $\lim_{n\to \infty}(1 + \frac{1}{n})^n$, and asked to solve, I would do so as follows: $$\lim_{n\to \infty}(1 + \frac{1}{n})^n$$ $$=(1 + \frac{1}{\infty})^\infty$$ $$=(1 + 0)^\infty$$ $$=1^\infty$$ $$=1$$ I’m aware that this limit is meant to equal to $e$, and so I ask: why don’t I get $e$ when I solve $\lim_{n\to \infty}(1 […]

Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$?

I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old. After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. […]

Solve a seemingly simple limit $\lim_{n\to\infty}\left(\frac{n-2}n\right)^{n^2}$

$$\lim_{n\to\infty}\left(\frac{n-2}n\right)^\left(n^2\right)$$ Why does this go to 0? Why can I not just divide each item in the fraction by n and assume it would go to 1?

Why is $0^0$ also known as indeterminate?

This question already has an answer here: Zero to the zero power – is $0^0=1$? 24 answers

Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?

The limit is $$\lim_{x\to0}\left(\frac{1}{\sin^2x}-\frac{1}{x^2}\right)$$ which I’m aware can be rearranged to obtain the indeterminate $\dfrac{0}{0}$, but in an attempt to avoid L’Hopital’s rule (just for fun) I tried using the fact that $\sin x\approx x$ near $x=0$. However, the actual limit is $\dfrac{1}{3}$, not $0$. In this similar limit, the approximation reasoning works out.

Find $\lim_{n \to \infty} \left$ (a question asked at trivia)

My friend’s trivia league had this math question: $$\lim_{n \to \infty} \left[\frac{(n+1)^{n + 1}}{n^n} – \frac{n^{n}}{(n-1)^{n-1}} \right]$$ After computing a few values, one could guess the answer is $e$ = 2.718…But how can we prove that is the limit? Someone offered up a hand-wavy proof like this: \begin{align} \lim_{n \to \infty} \left[\frac{(n+1)^{n + 1}}{n^n} – […]

Zero to the zero power – is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 \cdot 0^x = 1 \cdot 0^x$, so $0^0 = 1$ $0^0 = 0^x/0^x = 0/0$, which is undefined PS. I’ve read the […]

Question about the derivative definition

The derivative at a point $x$ is defined as: $\lim\limits_{h\to0} \frac{f(x+h) – f(x)}h$ But if $h\to0$, wouldn’t that mean: $\frac{f(x+0) – f(x)}0 = \frac0{0}$ which is undefined?

Why is $1^{\infty}$ considered to be an indeterminate form

From Wikipedia: In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is […]