Change of variable

Observation — The change of variable

In many limits the decisive trick is a change of variable: if $x\to x_0$ and the expression involves a quantity $g(x)\to 0$, one sets $t = g(x)$ and rewrites everything as a function of $t\to 0$, reducing to a standard limit.

Example

$\lim_{x\to 0}\frac{\sin(x^2)}{x^2}.$ We set $t = x^2$, with $t\to 0$: the expression becomes $\dfrac{\sin t}{t}\to 1$.

The substitution turns an unusual-looking limit into a standard limit already known: it is often the first thing to try when one recognises, “nested” in the expression, an infinitesimal quantity.

Links

Topics: Limits
Concepts: Change of variable · Standard limits
Skills: Calculating limits