Solution lim x → 2 x 2 2 = 4 and the one-sided

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Unformatted text preview: Solution. lim x → 2 x 2 + 2 = 4 and the one-sided limits of 1 x- 2 are ∞ from the left and ∞ from the right. From this the statement follows. Practice problems 1. Write out the definition of the following statements using inequalities: (a) lim x → 2 + g ( x ) = ∞ (b) lim x →∞ h ( x ) =-∞ 2. Find the following limits (you may use the limit laws) (a) lim x → 2 1 ( x- 2) 2 (b) lim x → 4- x 4- x (c) lim x →-∞ 1 x 3 3. Assume that lim x → 2- f ( x ) = ∞ and lim x → 2- g ( x ) = 5. Show (with ‘ ε- δ ’ proofs) that (a) lim x → 2- f ( x ) 2 = ∞ (b) lim x → 2- f ( x ) g ( x ) = ∞ (c) lim x → 2- g ( x )- f ( x ) = ∞ 2 (d) lim x → 2- g ( x ) /f ( x ) = 0. 4. Give examples of functions f ( x ), g ( x ) for which we have lim x → f ( x ) = ∞ , lim x → g ( x ) = ∞ and (a) lim x → f ( x )- g ( x ) = 5 (b) lim x → f ( x )- g ( x ) = ∞ (c) lim x → f ( x )- g ( x ) =-∞ (d) lim x → f ( x )- g ( x ) does not exists. 5. Show that for any non-constant polynomial p ( x ) the limit of p ( x ) will be ∞ or-∞ ad x → ∞ or x → -∞ . 6. Show that a rational function will have a finite nonzero limit at infinity if the degrees of the denominator and numerator are the same, and it will have a limit equal to zero if the degree of the numerator is smaller than the degree of the denominator. 7. Find the following limits (you may use the limit laws) (a) lim x → 1- x x 3- 1 (b) lim x → 3 + x 2- 2 x 3- x (c) lim x →-∞ x 2 x 3 +3 x- 2 (d) lim x →∞ x 2- 3 x √ x 4 +5 x- 2 (e) lim x →∞ √ x- 2- √ x- 4 Hint: You cannot use the limit laws immediately! 3...
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Solution lim x → 2 x 2 2 = 4 and the one-sided limits of...

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