극한·미분 심화 전체 문제

Explain what the indeterminate form “$1^\infty$” means.

Fill in the blanks:

The Quotient Rule is applied to $\displaystyle \frac{f(x)}{g(x)}$ when taking _____;

l'Hopital's Rule is applied to $\displaystyle \frac{f(x)}{g(x)}$ when taking certain _____.

Create (but do not evaluate!) a limit that returns “$\infty^0$”.

Create a function $f(x)$ such that $\displaystyle \lim_{x\to 1}f(x)$ returns “$0^0$”.

Create a function $f(x)$ such that $\displaystyle \lim_{x\to \infty}f(x)$ returns “$0\cdot\infty$”.

The definite integral was defined with what two stipulations?

If $\displaystyle \lim_{b\to \infty} \int_0^b f(x)\ dx$ exists, then the integral $\displaystyle \int_0^\infty f(x)\ dx$ is said to _____.

If $\displaystyle \int_1^\infty f(x)\ dx=10$, and $0\leq g(x)\leq f(x)$ for all $x$, then we know that $\displaystyle \int_1^\infty g(x)\ dx$ _____.

For what values of $p$ will $\displaystyle \int_1^\infty \frac1{x^p}\ dx$ converge?

For what values of $p$ will $\displaystyle \int_{10}^\infty \frac1{x^p}\ dx$ converge?

For what values of $p$ will $\displaystyle \int_{0}^1 \frac1{x^p}\ dx$ converge?

A particle starts moving at time $t=10$, and it bobs up and down, so that its height at time $t \geq 10$ is given by $\cos t$. True or false: the particle has height 1 at time $t=\arccos(1)$.