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Limits Formulas:

DEFINITION OF LIMITS
www.faastop.com الله π www.faastop.com الله π If the functional values of f(x)in the neighborhood of "𝑎" approach,a fixed real value k then we saythe limit of f(x) is k as x tends to a. limx→𝑎 𝑓(x) = K (or) Ltx→𝑎 𝑓(x) = K www.faastop.com الله π If the functional values of f(x)in the neighborhood of "𝑎" approach,a fixed real value k then we saythe limit of f(x) is k as x tends to a. If the functional values of f(x)in the neighborhood of "𝑎" approach,a fixed real value k then we saythe limit of f(x) is k as x tends to a. limx→𝑎 𝑓(x) = K (or) Ltx→𝑎 𝑓(x) = K limx→𝑎 𝑓(x) = K (or) If the functional values of f(x)in the neighborhood of "𝑎" approach,a fixed real value k then we saythe limit of f(x) is k as x tends to a. limx→𝑎 𝑓(x) = K (or) Ltx→𝑎 𝑓(x) = K Ltx→𝑎 𝑓(x) = K If the functional values of f(x)in the neighborhood of "𝑎" approach,a fixed real value k then we saythe limit of f(x) is k as x tends to a. limx→𝑎 𝑓(x) = K (or) Ltx→𝑎 𝑓(x) = K
STANDARD LIMITS
limx→𝑎 ——xⁿ-𝑎ⁿx-𝑎   = 𝑛𝑎ⁿ⁻¹www.faastop.com الله π limx→𝑎 ——xⁿ-𝑎ⁿx-𝑎   = 𝑛𝑎ⁿ⁻¹limx→𝑎 ——xⁿ-𝑎ⁿx-𝑎   = 𝑛𝑎ⁿ⁻¹
limx→0 ——𝑠𝑖𝑛xx   = 1www.faastop.com الله π limx→0 ——𝑠𝑖𝑛xx   = 1limx→0 ——𝑠𝑖𝑛xx   = 1
limx→∞ ——𝑠𝑖𝑛xx   = 0www.faastop.com الله π limx→∞ ——𝑠𝑖𝑛xx   = 0limx→∞ ——𝑠𝑖𝑛xx   = 0
limx→0 ——𝑡𝑎𝑛xx   = 1www.faastop.com الله π limx→0 ——𝑡𝑎𝑛xx   = 1limx→0 ——𝑡𝑎𝑛xx   = 1
limx→0 ——𝑒x      -1x   = 1www.faastop.com الله π limx→0 ——𝑒x      -1x   = 1limx→0 ——𝑒x      -1x   = 1
limx→0 ——In(1+x)x    =1www.faastop.com الله π limx→0 ——In(1+x)x    =1limx→0 ——In(1+x)x    =1
limx→∞ (1+x)1/x = 𝑒www.faastop.com الله π limx→∞ (1+x)1/x = 𝑒limx→∞ (1+x)1/x = 𝑒
limx→0 (1+𝑎x)1/x = 𝑒𝑎www.faastop.com الله π limx→0 (1+𝑎x)1/x = 𝑒𝑎limx→0 (1+𝑎x)1/x = 𝑒𝑎
limx→∞[1+ ——1x     ]                                   x = 𝑒www.faastop.com الله π limx→∞[1+ ——1x     ]                                   x = 𝑒limx→∞[1+ ——1x     ]                                   x = 𝑒
limx→∞[1+ ——𝑎x     ]                                   x = 𝑒𝑎www.faastop.com الله π limx→∞[1+ ——𝑎x     ]                                   x = 𝑒𝑎limx→∞[1+ ——𝑎x     ]                                   x = 𝑒𝑎
limx→∞ (x)1/x = 1www.faastop.com الله π limx→∞ (x)1/x = 1limx→∞ (x)1/x = 1
limx→0 (x)x = 1www.faastop.com الله π limx→0 (x)x = 1limx→0 (x)x = 1
INDETERMINATE FORMS
[ ——00,  ——∞∞, 0×∞, ∞-∞, 1∞, 0∞, ∞0 ]www.faastop.com الله π [ ——00,  ——∞∞, 0×∞, ∞-∞, 1∞, 0∞, ∞0 ][ ——00,  ——∞∞, 0×∞, ∞-∞, 1∞, 0∞, ∞0 ]
PROPERTIES OF LIMITS
Assume Ltx→𝑎𝑓(x) and Ltx→𝑎g(x) both exist and L is any real number, then www.faastop.com الله π Assume Ltx→𝑎𝑓(x) and Ltx→𝑎g(x) both exist and L is any real number, then Assume Ltx→𝑎𝑓(x) and Ltx→𝑎g(x) both exist and L is any real number, then
Ltx→𝑎L𝑓(x) =LLtx→𝑎𝑓(x) www.faastop.com الله π Ltx→𝑎L𝑓(x) =LLtx→𝑎𝑓(x) Ltx→𝑎L𝑓(x) =LLtx→𝑎𝑓(x)
limx→𝑎 [𝑓(x)+₋ 𝑔(x)] = limx→𝑎 𝑓(x)+₋ limx→𝑎 𝑔(x)www.faastop.com الله π limx→𝑎 [𝑓(x)+₋ 𝑔(x)] = limx→𝑎 𝑓(x)+₋ limx→𝑎 𝑔(x)limx→𝑎 [𝑓(x)+₋ 𝑔(x)] = limx→𝑎 𝑓(x)+₋ limx→𝑎 𝑔(x)
limx→𝑎 [𝑓(x) 𝑔(x)] = limx→𝑎 𝑓(x) limx→𝑎 𝑔(x)www.faastop.com الله π limx→𝑎 [𝑓(x) 𝑔(x)] = limx→𝑎 𝑓(x) limx→𝑎 𝑔(x)limx→𝑎 [𝑓(x) 𝑔(x)] = limx→𝑎 𝑓(x) limx→𝑎 𝑔(x)
limx→𝑎[𝑓(x)𝑔(x)] = limx→𝑎𝑓(x) limx→𝑎𝑔(x) ,  where limx→𝑎𝑔(x) ≠ 0www.faastop.com الله π limx→𝑎[𝑓(x)𝑔(x)] = limx→𝑎𝑓(x) limx→𝑎𝑔(x) ,  where limx→𝑎𝑔(x) ≠ 0limx→𝑎[𝑓(x)𝑔(x)] = limx→𝑎𝑓(x) limx→𝑎𝑔(x) ,  where limx→𝑎𝑔(x) ≠ 0
limx→𝑎[𝑓(x)]ⁿ =[limx→𝑎 𝑓(x)]ⁿ www.faastop.com الله π limx→𝑎[𝑓(x)]ⁿ =[limx→𝑎 𝑓(x)]ⁿ limx→𝑎[𝑓(x)]ⁿ =[limx→𝑎 𝑓(x)]ⁿ
limx→𝑎 ⁿ  𝑓(x) = ⁿ limx→𝑎 𝑓(x) www.faastop.com الله π limx→𝑎 ⁿ  𝑓(x) = ⁿ limx→𝑎 𝑓(x) limx→𝑎 ⁿ  𝑓(x) = ⁿ limx→𝑎 𝑓(x)
CONTINUITYT
www.faastop.com الله π   A function f(x) is said to be continuous at x = 𝑎 if ,   Ltx→𝑎⁻𝑓(x) = Ltx→𝑎⁺𝑓(x) = f(a)www.faastop.com الله π   A function f(x) is said to be continuous at x = 𝑎 if ,   A function f(x) is said to be continuous at x = 𝑎 if ,   Ltx→𝑎⁻𝑓(x) = Ltx→𝑎⁺𝑓(x) = f(a)   Ltx→𝑎⁻𝑓(x) = Ltx→𝑎⁺𝑓(x) = f(a)   A function f(x) is said to be continuous at x = 𝑎 if ,   Ltx→𝑎⁻𝑓(x) = Ltx→𝑎⁺𝑓(x) = f(a)

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