Form :- (→0/→0) => indeterminate form.
Method 1 :- Use taylor series formula for e$^{x}$ and cos(x).
e$^{x}$ = 1 + x + x$^{2}$/2! + x$^{3}$/3! + x$^{4}$/4! + ….
=> e$^{x}$ – 1 = x + x$^{2}$/2! + x$^{3}$/3! + x$^{4}$/4! + …. = x( 1 + x/2! + x$^{2}$/3! + x$^{3}$/4! + ….)
=> x(e$^{x}$ – 1) = x$^{2}$( 1 + x/2! + x$^{2}$/3! + x$^{3}$/4! + ….)
cos(x) = 1 – x$^{2}$/2! + x$^{4}$/4! – x$^{6}$/6! + x$^{8}$/8! – ….
=> (cos(x) – 1) = – x$^{2}$/2! + x$^{4}$/4! – x$^{6}$/6! + …. = –x$^{2}$(1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)
$\lim_{x→0}$[x(e$^{x}$ – 1) + 2(cos(x) – 1)] / x(1 – cos(x))
= $\lim_{x→0}$[x$^{2}$( 1 + x/2! + x$^{2}$/3! + x$^{3}$/4! +….) + –2x$^{2}$(1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)] / x$^{3}$(1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)
= $\lim_{x→0}$[( 1 + x/2! + x$^{2}$/3! + x$^{3}$/4! +….) + –2(1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)] / x(1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)
= $\lim_{x→0}$[( 1 + x/2! + x$^{2}$/3! + x$^{3}$/4! +….) + (-1 + 2x$^{2}$/4! - 2x$^{4}$/6! + ….)] / x(1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)
= $\lim_{x→0}$[(x/2! + x$^{2}$/3! + x$^{3}$/4! +….) + (2x$^{2}$/4! - 2x$^{4}$/6! + ….)] / x(1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)
= $\lim_{x→0}$[(1/2! + x/3! + x$^{2}$/4! +….) + (2x/4! - 2x$^{3}$/6! + ….)] / (1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)
= $\lim_{x→0}$[(1/2! + x/3! + x$^{2}$/4! +….) + (2x/4! - 2x$^{3}$/6! + ….)] / (1/2! – x$^{2}$/4! + x$^{4}$/6! – ….)
= $\lim_{x→0}$(1/2)/(1/2) = 1
Method 2 :- L'Hopital's Rule(apply only for →0/→0 or →$\infty$/→$\infty$ indeterminate form)
$\lim_{x→0}$[x(e$^{x}$ – 1) + 2(cos(x) – 1)] / x(1 – cos(x))
= $\lim_{x→0}$[(e$^{x}$ – 1) + xe$^{x}$ - 2sin(x)] / [(1-cos(x)) + x sin(x)] (→0/→0)
= $\lim_{x→0}$[2e$^{x}$ + xe$^{x}$ - 2cos(x)] / [2sin(x) + x cos(x)] (→0/→0)
= $\lim_{x→0}$[3e$^{x}$ + xe$^{x}$ + 2sin(x)] / [3cos(x) - x sin(x)] (not →0/→0)
= $\lim_{x→0}$(3)/(3) = 1