Quick questions on Maclaurin series explained: H2 Further Mathematics

The Maclaurin series expands a function as a power series about $x = 0$:

When derivatives are easy, differentiate $\mathrm{f}$ repeatedly, evaluate each derivative at $0$, and substitute. For a function defined implicitly (for example $y = \ln(1 + \sin x)$), it is usually neater to clear the denominator to get a relation such as $(1 + \sin x)y' = \cos x$, then differentiate that relation repeatedly with the product rule, evaluating at $x = 0$ at each stage to generate $y', y'', y''', \dots$ in turn.

Build new series by substitution (for example $x \to 2x$), multiplication, or term-by-term differentiation and integration. This is faster than repeated differentiation when a standard series applies.

A limit of the form $\dfrac{0}{0}$ as $x \to 0$ is found by replacing numerator and denominator with their series and cancelling the lowest powers of $x$. The surviving constant term is the limit. This is a clean alternative to repeated L'Hopital differentiation.

Write the Maclaurin series of $\mathrm{e}^x$ up to the term in $x^3$. [1 mark]

Use series to evaluate $\displaystyle\lim_{x\to 0}\dfrac{1 - \cos x}{x^2}$. [2 marks]

If $y = (1 + x)^{1/2}$, give the first three terms of its Maclaurin series. [2 marks]