IB Mathematics Analysis and Approaches Question Type 2: Using Tables of Values of Small Increments to Explore Different Functions Exercises

Evaluate $k(x)=\frac{\sin x}{x}$ at $x=0.1$, $0.01$, $0.001$, $0.0001$ using a table, and estimate ${\lim }_{x\to 0}\frac{\sin x}{x}$.

Construct a table of values for $f(x)=\frac{1}{x}$ at $x=1$, $0.1$, $0.01$, $0.001$. Compute the values and describe the trend as $x$ approaches 0 from the positive side.

Construct a table for $q(x)=\frac{2^x-1}{x}$ at $x=0.1$, $0.01$, $0.001$, $0.0001$. Use it to estimate ${\lim }_{x\to 0}\frac{2^x-1}{x}$. [3 marks]

Construct a table of values for $f(x)$ at $x=0.1$, $0.01$, $0.001$, and $0.0001$. Use the table to estimate ${\lim }_{x\to 0}\frac{e^x-1-x}{x^2}$.

Construct a table for $m(x)=\frac{\tan x}{x}$ at $x=0.1$, $0.01$, $0.001$, $0.0001$ and use it to estimate ${\lim }_{x\to 0}\frac{\tan x}{x}$.

Using a table of values at $x=0.1$, $0.01$, $0.001$, $0.0001$, evaluate $g(x)=\frac{1}{1+x}$. Describe the behavior of $g(x)$ as $x\to 0$.

Using a table of values at $x=0.1$, $0.01$, $0.001$, $0.0001$, evaluate $p(x)=(1+2x{)}^{1/x}$ and estimate ${\lim }_{x\to 0}(1+2x{)}^{1/x}$.

Fill in a table for $h(x)=(1+x{)}^{1/x}$ at $x=0.1$, $0.01$, $0.001$, $0.0001$. Use your results to estimate ${\lim }_{x\to 0}(1+x{)}^{1/x}$.

Use a table of values at $x=0.1$, $0.01$, $0.001$, $0.0001$ to compute $f(x)=\frac{1-\cos x}{x^2}$ and estimate ${\lim }_{x\to 0}\frac{1-\cos x}{x^2}$.

Use a table with $x=0.1,0.01,0.001,0.0001$ to evaluate $i(x)=\frac{\ln (1+x)}{x}$. Estimate the limit as $x\to 0$.

Construct a table for $j(x)=\frac{e^x-1}{x}$ at $x=0.1$, $0.01$, $0.001$, $0.0001$. Use it to approximate ${\lim }_{x\to 0}\frac{e^x-1}{x}$.