Class 11 Maths Limits Derivatives Theorems

Theorem 1: Refer ExamFear video lessons for Proof.

Numerical: Evaluate Lim xà 1   [ (x15 -1)/ (x10 -1)]

Solution: Rewrite the expression as

Lim xà 1   [ (x15 -1)/ (x -1)] *  Lim xà 1   [ (x -1)/ (x10 -1)]

= 15 (1)14 / 10 (1)9

= 3/2

Theorem 2: Let f and g be two real valued functions with the same domain such that f (x) ≤ g( x) for all x in the domain of definition, For some a,  if both lim x→a f(x) and lim x→a g(x) exist, then  lim x→a f(x) ≤ lim x→a g(x).

Refer ExamFear video lessons for Proof.

Sandwich Theorem 3: Let f, g and h be real functions such that f (x) ≤ g( x) ≤ h(x) for all x in the common domain of definition. For some real number a, if lim x→a f(x) = l = lim x→a h(x), then lim x→a g(x) = l.

Theorem 4:  lim x→0 (sin x)/ x = 1

Theorem 5:  lim x→0 (1 - cos x)/ x = 0

Refer ExamFear video lessons for Proof.

Numerical:  Evaluate lim x→0 (sin 5x)/(Sin 4x)

Solution: We can rewrite the expression as  lim x→0 (sin 5x)/x)  *  lim x→0  x/(Sin 4x)

= 5/4 * lim x→0 (sin 5x)/5x)  / lim x→0  (Sin 4x)/ 4

= 5/4 * 1 * 1

= 5/4

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