1. Chapter 13 Class 11 Limits and Derivatives
2. Serial order wise

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Ex 13.1, 25 (Method 1) Evaluate lim﷮x→0﷯ f(x), where f(x) = x﷯﷮x﷯﷮0,﷯﷯, x≠0﷮x=0﷯ Finding limit at x = 0 lim﷮x→ 0﷮−﷯﷯ f(x) = lim﷮x→ 0﷮+﷯﷯ f(x) = lim﷮x→0﷯f(x) Thus, lim﷮ x→0﷮−﷯﷯f(x) = –1 & lim﷮ x→0﷮+﷯﷯f(x) = 1 Since –1 ≠ 1 So, 𝒍𝒊𝒎﷮ 𝐱→𝟎﷮−﷯﷯ f(x) ≠ 𝒍𝒊𝒎﷮ 𝐱→𝟎﷮+﷯﷯ f(x) So, left hand limit & right hand limit are not equal Hence, 𝒍𝒊𝒎﷮𝐱→𝟎﷯ f(x) does not exist Ex13.1, 25 (Method 2) Evaluate lim﷮x→0﷯ f(x), where f(x) = x﷯﷮x﷯﷮0,﷯﷯, x≠0﷮x=0﷯ We know that lim﷮x→a﷯ f(x) exist only if lim﷮ x→𝑎﷮−﷯﷯f(x) = lim﷮ x→𝑎﷮+﷯﷯f(x) Similarly in this question we have find limits First we have to prove limit exists by proving lim﷮ x→0﷮−﷯﷯f(x) = lim﷮ x→0﷮+﷯﷯f(x) For 𝒍𝒊𝒎﷮ 𝐱→𝟎﷮−﷯﷯ f(x) f(x) = x﷯﷮x﷯ So, as x tends to 0, f(x) tends to –1 ∴ 𝑙𝑖𝑚﷮ 𝑥→0﷮−﷯﷯ f(x) = –1 For 𝒍𝒊𝒎﷮ 𝐱→𝟎﷮+﷯﷯ f(x) f(x) = x﷯﷮x﷯ So, as x tends to 0, f(x) tends to 1 ∴ 𝑙𝑖𝑚﷮ 𝑥→0﷮+﷯﷯ f(x) = 1 Thus, lim﷮ x→0﷮−﷯﷯f(x) = –1 & lim﷮ x→0﷮+﷯﷯f(x) = 1 Since –1 ≠ 1 So, lim﷮ x→0﷮−﷯﷯f(x) ≠ lim﷮ x→0﷮+﷯﷯f(x) So, left hand limit & right hand limit are not equal Hence, 𝒍𝒊𝒎﷮𝐱→𝟎﷯ f(x) does not exist