IB Mathematics Applications & Interpretation Question Type 2: Using Tables of Values of Small Increments to Explore Different Functions Exercises

Evaluate the following one-sided limits: $$\underset{x\to 2^{-}}{\lim }\frac{x-2}{|x-2|}\text{and}\underset{x\to 2^{+}}{\lim }\frac{x-2}{|x-2|}.$$

Evaluate the two-sided behaviour of $$\underset{x\to 0^{+}}{\lim }{e}^{\frac{1}{x}}\text{and}\underset{x\to 0^{-}}{\lim }{e}^{\frac{1}{x}}$$

Find the following limits:

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

Evaluate the one-sided limits $$\underset{x\to 2^{-}}{\lim }\frac{x^2-4}{x-2}\text{and}\underset{x\to 2^{+}}{\lim }\frac{x^2-4}{x-2}.$$

Find the behavior of the following limits: $$\underset{x\to 0^{-}}{\lim }\sin \left(\frac{1}{x}\right)\text{and}\underset{x\to 0^{+}}{\lim }\sin \left(\frac{1}{x}\right)$$

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 $$\underset{x\to 0^{+}}{\lim }x\sin (1/x)\text{and}\underset{x\to 0^{-}}{\lim }x\sin (1/x).$$

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

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

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