Question Type 2: Using tables of values of small increments to explore different functions Exercises

Evaluate the following one-sided limits: $\lim_{x\to2^-}\frac{x-2}{|x-2|}\quad\text{and}\quad\lim_{x\to2^+}\frac{x-2}{|x-2|}.$

Evaluate the two-sided behaviour of $\lim_{x\to0^+}e^{\frac{1}{x}}\quad\text{and}\quad\lim_{x\to0^-}e^{\frac{1}{x}}$

Find the following limits:

$\lim_{x \to 3^-} \frac{1}{(x - 3)^2} \quad \text{and} \quad \lim_{x \to 3^+} \frac{1}{(x - 3)^2}$

Evaluate the one-sided limits $\lim_{x\to2^-}\frac{x^2-4}{x-2}\quad\text{and}\quad\lim_{x\to2^+}\frac{x^2-4}{x-2}.$

Find the behavior of the following limits: $\lim_{x\to0^-} \sin\left(\frac{1}{x}\right) \quad \text{and} \quad \lim_{x\to0^+} \sin\left(\frac{1}{x}\right)$

Determine the one-sided limits of the natural logarithm function: $\lim_{x \to 0^+} \ln(x)$ and $\lim_{x \to 0^-} \ln(x)$.

Determine $\lim_{x \to 1^-} \frac{|x - 1|}{x - 1}$ and $\lim_{x \to 1^+} \frac{|x - 1|}{x - 1}$.

Evaluate the one-sided limits of $\lim_{x\to0^+}x\sin\bigl(1/x\bigr) \quad \text{and} \quad \lim_{x\to0^-}x\sin\bigl(1/x\bigr).$

Evaluate the limits $\lim_{x \to 0^+} \frac{1}{x} \quad \text{and} \quad \lim_{x \to 0^-} \frac{1}{x}$

Consider the piecewise function $f(x) = \begin{cases} x^2, & x < 1 \\ 2x - 1, & x \ge 1. \end{cases}$ Find the values of $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$.

Evaluate the limits $\lim_{x\to\pi^-}\tan(x)\quad\text{and}\quad\lim_{x\to\pi^+}\tan(x).$