Calculus of one variable

We want $|x^2 - 9| < \varepsilon$ whenever $|x - 3| < \delta$. Following the lesson's recipe $\delta = \min\big(1, \varepsilon/(2|a| + 1)\big)$ with $a = 3$, which $\delta$ does it give for $\varepsilon = 0.7$?

Compute $\displaystyle\lim_{x \to 0} \frac{1 - \cos(4x)}{x^2}$.

For $f(x) = |x|$ on $[-1, 2]$, the chord slope is $\frac{f(2) - f(-1)}{2 - (-1)} = \frac13$, but $f'(x) \in \{-1, 1\}$ wherever it exists. Which hypothesis of the mean value theorem fails?

Compute $\displaystyle\int_1^e \ln x\,dx$.

Which of these series converge? (Select all that apply.)

Using the Lagrange bound $|e - T_n(1)| \le \dfrac{e}{(n + 1)!}$ for the Maclaurin polynomial of $e^x$ at $x = 1$, what is the smallest degree $n$ that guarantees $|e - T_n(1)| < 10^{-6}$? (Use $e < 3$ if you like; the answer is the same.)

A student writes: $\displaystyle\lim_{x \to 0} \frac{x^2 + 1}{x + 1} = \lim_{x \to 0} \frac{2x}{1} = 0$. What is wrong?

Let $f(x) = e^{-1/x^2}$ for $x \ne 0$ and $f(0) = 0$. On which set does $f$ equal the sum of its Maclaurin series?

Which Lean statement correctly formalises the mean value theorem? (Here Icc a b is $[a, b]$ and Ioo a b is $(a, b)$.)

Write exp_taylor(x, n) returning $T_n(x) = \sum_{k=0}^{n} x^k/k!$, and degree_needed(x, tol) returning the least $n \ge 0$ for which the Lagrange bound $e^{\max(x, 0)}\,|x|^{n+1}/(n+1)!$ is less than tol. Then check that $T_n(x)$ really is within tol of math.exp(x).