V limits and continuity the function fx has limit l as

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Unformatted text preview: x satisfying 0 < |x – a|< δ. Evaluate lim[ f (x ) ± g ( x )] = lim f ( x ) ± lim g ( x ) , SOLUTION: Let L be the limit. lim[ f ( x ) ⋅ g ( x )] = lim f ( x ) ⋅ lim g ( x ) , ⎛ x 2 − 5 − 1 + x ⎞⎛ x 2 − 5 + 1 + x ⎞ ⎜ ⎟⎜ ⎟ ⎠⎝ ⎠ L = lim ⎝ ⎛ x2 − 5 + 1 + x ⎞ 2 x →3 x −9⎜ ⎟ ⎝ ⎠ x2 − 9 x →3 d 1 du d u du , , (ln u) = ⋅ (e ) = e u du u dx dx dx du du d , (a ) = a u (ln a ) (sin x ) = cos x , dx dx dx d d (cos x ) = − sin x , (tan x ) = sec 2 x , dx dx d 2 (cot x ) = − csc x , dx More free study sheet and practice tests at: x→a x→a x→a x→a x→a ⎡ f ( x ) ⎤ lim f ( x ) , = x→a lim ⎢ x→a g (x ) ⎥ ⎦ lim g ( x ) ⎣ x→a [ n x→a x→a lim[ f ( x )] = lim f ( x ) x→a x 1+x⎞ ⎟ ⎠ 5 = 24 1+x⎞ ⎟ ⎠ h →0 2 lim ln 3 + 2ln 5 SOLUTION: e 0 form. Therefore take 0 3x 2 + 6 x − 10 14 = =7 x→2 3x 2 − 6 x + 2 2 EXAMPLE: Simplify e f ′(x ) = lim x − 3x + 2 x Note that this limit is in SOLUTION: + 3 )⎛ ⎜ ⎝ (x + 2 )(x − 3 ) − 3 )⎛ x 2 − 5 + ⎜ ⎝ (x + 2 ) DIFFERENTIATION BY FIRST PRINCIPLES: 2 EXAMPLE: )(x The Derivative x 3 + 3x 2 − 10 x . lim +3 (x = lim SOLUTION: Evaluate log8 2 + log2 8. (x x→ 3 f ( x) f ′( x ) = lim x → a g ′( x ) g ( x) x →2 log b ( x n ) = n ⋅ log b ( x) log 8 8 = lim n ( = lim x...
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