Limits of functions and sequences, continuity, and standard limit techniques.
These practice problems are the opening of calculus: decide whether a limit exists, and compute it when it does. The set covers sequences, functions of a real variable, one-sided limits, limits at infinity, and continuity. It matches the first weeks of a calculus course, before the derivative is defined. Each problem has a full step-by-step solution, so you can try the argument yourself and then compare it with the write-up.
Start with Standard Limits. That chapter holds algebraic limits, limits of rational expressions, and the standard trigonometric and exponential limits that show up again inside derivatives. Piecewise Functions and Parameters asks you to compare one-sided limits, choose a constant so that a limit exists, and read a limit from a formula that changes at a point. Continuity asks where a function is continuous, and how a removable discontinuity can be filled so the function becomes continuous there.
Use the difficulty control to begin with Easy problems, which are single standard limits, and then move to Hard problems that combine several devices in one expression. When the computations feel routine, continue with Differential Calculus, where these limits become the definition of the derivative. If derivatives are already comfortable, the next large collection is Integrals.
Problem 1
HardStandard Limits
Evaluate the limit:
$$
L = \lim_{n \to \infty} \left( \frac{n+2}{n+1} \right)^{n+3}.
$$
For a fixed positive integer $n$, evaluate the limit:
$$
L = \lim_{x \to +\infty} \frac{(x+1)(x^2+1)\cdots(x^n+1)}{\left((nx)^n+1\right)^{\frac{n+1}{2}}}.
$$
Consider the piecewise function
$$
f(x,\alpha) =
\begin{cases}
\alpha k + 1, & \text{if } x < 1, \\
\sin(x\pi) + 2, & \text{if } x > 1,
\end{cases}
$$
where $\alpha$ and $k$ are constants. Analyze the behavior of $f$ on each piece and determine the condition on $\alpha$ and $k$ under which the two sides can be joined continuously at $x=1$.
Given the function
$$
f(x) = \frac{1}{1+2^{1/x}},
$$
find $\displaystyle\lim_{x\to0^+} f(x)$, $\displaystyle\lim_{x\to0^-} f(x)$, and $\displaystyle\lim_{x\to+\infty} f(x)$.
Given the function $f(x) = \sin\left(\dfrac{1}{x}\right)$, find $\displaystyle\lim_{x\to+\infty} f(x)$, and show that $\displaystyle\lim_{x\to0^+} f(x)$ and $\displaystyle\lim_{x\to0^-} f(x)$ do not exist.
Given the piecewise function
$$
f(x,\alpha) =
\begin{cases}
\alpha x^2 + x + 3, & \text{if } x \le 1, \\
\sin\left(\dfrac{\pi x}{2}\right) + 2\alpha, & \text{if } x > 1,
\end{cases}
$$
determine the value of $\alpha$ for which $f$ is continuous at $x=1$.
Let
$$
f(x) = x^{n-1}\sin\left(\frac1x\right), \quad x \neq 0, \qquad f(0) = 0.
$$
Determine the values of $n$ for which $f$ is continuous at $x=0$, and the values of $n$ for which $f$ is differentiable at $x=0$.
Let
$$
f(x) = \ln\left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}, \quad x \neq 0, \qquad f(0) = \alpha.
$$
Determine the value of $\alpha$ for which $f$ is continuous at $x=0$.
Evaluate the limit, and determine how its value depends on the positive parameter $n$:
$$
L(n) = \lim_{x \to 0^+} \frac{\sqrt[n]{1+x\sin(3x)} \cdot \sqrt[n]{\tan(2x)-1+1}}{e^{\sin x}-1} = \lim_{x \to 0^+} \frac{\sqrt[n]{1+x\sin(3x)} \cdot \sqrt[n]{\,1+\left(\tan(2x)-1\right)\,}}{e^{\sin x}-1}.
$$
(Note: the second radical simplifies since $1+\left(\tan(2x)-1\right)=\tan(2x)$.)
Evaluate the limit:
$$
L = \lim_{x \to 0} \frac{\cos(xe^x)-\cos(xe^{-x})}{x^3},
$$
using the sum-to-product identity
$$
\cos(xe^x)-\cos(xe^{-x}) = -2\sin\left(\frac{xe^x+xe^{-x}}{2}\right)\sin\left(\frac{xe^x-xe^{-x}}{2}\right).
$$
Given the piecewise function
$$
f(x) =
\begin{cases}
kx^2+1, & x<2, \\
5, & x=2, \\
x+3, & x>2,
\end{cases}
$$
determine the value of $k$ (if any) for which $f$ is continuous at $x=2$.