Introduction to Mathematical Analysis I - Second Edition

67 3.1.4 Let f : D → R and let ¯ x be a limit point of D . Suppose f ( x ) ≥ 0 for all x ∈ D . Prove that if lim x → ¯ x f ( x ) = ` , then lim x → ¯ x p f ( x ) = √ `. 3.1.5 Find lim x → 0 x sin ( 1 / x ) . 3.1.6 I Let f be the function given by f ( x ) = ( x , if x ∈ Q ∩ [ 0 , 1 ] ; 1 − x , if x ∈ Q c ∩ [ 0 , 1 ] . Determine which of the following limits exist. For those that exist find their values. (a) lim x → 1 / 2 f ( x ) . (b) lim x → 0 f ( x ) . (c) lim x → 1 f ( x ) . 3.2 LIMIT THEOREMS Here we state and prove various theorems that facilitate the computation of general limits. Definition 3.2.1 Let f , g : D → R and let c be a constant. The functions f + g , f g , and c f are respectively defined as functions from D to R by ( f + g )( x ) = f ( x )+ g ( x ) , ( f g )( x ) = f ( x ) g ( x ) , ( c f )( x ) = c f ( x ) for x ∈ D . Let e D = { x ∈ D : g ( x ) 6 = 0 } . The function f g is defined as a function from e D to R by f g ( x ) = f ( x ) g ( x ) for x ∈ e D . Theorem 3.2.1 Let f , g : D → R and let c ∈ R . Suppose ¯ x is a limit point of D and lim x → ¯ x f ( x ) = `, lim x → ¯ x g ( x ) = m . Then (a) lim x → ¯ x ( f + g )( x ) = ` + m , (b) lim x → ¯ x ( f g )( x ) = ` m , (c) lim x → ¯ x ( c f )( x ) = c ` , (d) lim x → ¯ x f g ( x ) = ` m provided that m 6 = 0.