Compute $$\lim_{x\to 4}\frac{3-\sqrt{x+5}}{x-4}$$
Is the $y=\frac{1}{x}$ a continuous function?
Show that $$\lim_{x\to 0}\ \frac{x}{\sin{x}}=1$$
Show the following sequence is convergent:
$$\frac{1}{1^2},\ \frac{1}{2^2},\ \frac{1}{3^2},\ \cdots,\ \frac{1}{n^2},\ \cdots$$
Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.
Show that $$\lim_{x\to 0}\frac{e^x-1}{x}=1$$
Find the value of
$$\lim_{x\to\infty}\frac{\sin{x}}{x}$$
Compute the derivative of $f(x)=x^n$.
Show that $$\frac{d}{dx} e^x = e^x$$
Given $\frac{d}{dx} e^x = e^x$, find the value of $\frac{d}{dx} \ln x$.
Find the derivative of function $y=\sin{x}$.
Find the derivative of $\arcsin{x}$.
Let $f(x)$ be an odd function which is differentiable over $(-\infty, +\infty)$. Show that $f'(x)$ is even.
Compute the limit of the power series below as a rational function in $x$:
$$1\cdot 2 + (2\cdot 3)x + (3\cdot 4)x^2 + (4\cdot 5)x^3 + (5\cdot 6)x^4+\cdots,\qquad (|x| < 1)$$
Compute $$1-\frac{1\times 2}{2}+\frac{2\times 3}{2^2}-\frac{3\times 4}{2^3}+\frac{4\times 5}{2^4}-\cdots$$
Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:
Find the coordinates of the center of mass of the $\frac{1}{4}$ disc defined by
$$\{(x, y) | x\ge 0, y\ge 0, x^2 + y^2 \le 1\}$$
assuming the density is uniform.
Compute $$I=\int \frac{x\cos{x}-\sin{x}}{x^2 + \sin^2{x}} dx$$
Find the maximum and minimal values of the function
$$f(x)=(x^2-4)^8 -128\sqrt{4-x^2}$$
over its domain.
Find all quadratic polynomials $p(x)=ax^2 + bx + c$ such that graphs of $p(x)$ and $p'(x)$ are tangent to each other at point $(2, 1)$.
Determine if the following infinite series is convergent or divergent:
$$\sum_{n=2}^{\infty}\frac{1}{(\ln n)^{\ln \ln n}}$$
Show that $\ln x < \sqrt{x}$ holds for all positive $x$.
Evaluate $$\int_{0}^{\pi}\frac{x\sin{x}}{1+\cos^2 x}dx$$
Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(x)+f(\frac{1}{x})$.