True or false? If lim_x→0 f(x) = ∞ and lim_x→0 g(x) = ∞, then lim_x→0 [f(x) - g(x)] = 0.

Solution:

If lim_x→0 f(x) = ∞ and lim_x→0 g(x) = ∞,

then lim_x→0 [f(x) - g(x)] = 0, is false.

lim_x→0 [f(x) - g(x)]

= lim_x→0 f(x) - lim_x→0 g(x)

= ∞ - ∞

This is in-determinant.

We say ∞ - ∞ as in-determinant because the first infinity may be the result of say of the form 4/0 whereas the second may be of the form say 8/0.

Since the numerators are different, 4/0 and 8/0 cannot be equal to the same infinity.

We have different types of in- determinant forms: 0/0, ∞/∞, 1∞ , 0⁰, etc,

which are usually observed in limit problems.

Hence, the given limit is a false statement.

True or false? If lim_x→0 f(x) = ∞ and lim_x→0 g(x) = ∞, then lim_x→0 [f(x) - g(x)] = 0.

Summary:

If lim_x→0 f(x) = ∞ and lim_x→0 g(x) = ∞, then lim_x→0 [f(x) - g(x)] = 0, is a false statement.