Calculus Cheat Sheet All Reduced Differential & Integral Calculus I (MATH 203) Concordia University

Calculus Cheat Sheet All Reduced Differential & Integral Calculus I (MATH 203) Concordia University

Precise Definition : We say limxafx L if for every 0 there is a 0such that whenever 0xa then fx L. “Working” Definition : We say limxafxL if we can make fx as close to L as we want by taking x sufficiently close to a (on either side of a) without letting xa. Right hand limit : limxafxL. This has the same definition as the limit except it requires xa. Left hand limit : limxafx L. This has the same definition as the limit except it requires xa. Limit at Infinity : We saylimxfx L  if we can make fx as close to L as we want by taking x large enough and positive. There is a similar definition for limxfxL  except we require x large and negative. Infinite Limit : We say limxafx if we can make fx arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting xa. There is a similar definition for limxafx except we make fx arbitrarily large and negative.Relationship between the limit and one-sided limits limxafxL   lim limxa xafxfxL  lim limxa xafxfxL  limxafxL  lim limxa xafx fx  limxafx Does Not Exist Properties Assume limxafx and limxagx both exist and c is any number then, 1.  lim limxa xacf x c f x 2.    lim lim limxa xa xafxgx fx gx  3.    lim lim limxa xa xafxgx f x gx 4. limlim limxaxaxafxfxgxgx provided lim 0xagx 5.  lim limnnxa xafxfx 6.  lim limnnxa xafx fxBasic Limit Evaluations at  Note : sgn 1a if 0a and sgn 1a if 0a.1. lim xx e & lim 0xx  e 2. lim lnxx  & 0lim lnxx 3. If 0rthen lim 0rxbx  4. If 0r and rxis real for negative x then lim 0rxbx  5. n even : lim nxx  6. n odd : lim nxx  & lim nxx  

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