Question

lim x tends to 0 sinx/ x=1​

Answer 1

Answer:

The limit lim

x→1

cos

x

2−1

x−1

can be evaluated using what we know

about the composition of continuous functions. Indeed, since cos is

continuous on R, we have

lim

x→1

cos(g(x)) = cos lim

x→1

g(x)

whenever lim

x→1

g(x) exists. Here g(x) = x

2 − 1

x − 1

= x + 1 for x 6= 1.

Therefore,

lim

x→1

cos

x

2 − 1

x − 1

= lim

x→1

cos (x + 1) = cos lim

x→1

(x + 1) = cos 2.