Limits and Continuity: IITJEE Maths : Tutorial problems 1

Problem 1: Which of the following is an indeterminate form ? (a) 1^{1} (b) 0^{1} (c) 1^{0} (d) 0^{0}

Problem 2: Which of the following is not an indeterminate form ? (a) 1^{1} (b) 0 \times \infty (c) 1^{\infty} (d) \infty^{0}

Problem 3: If \lim_{x \rightarrow c} f(x) and \lim_{x \rightarrow c}g(x) exists then which of the following conditions is not always correct ? (i) \lim_{x \rightarrow c}(f(x)+g(x)) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c}g(x) (ii) \lim_{x \rightarrow c}(f(x)-g(x)) = \lim_{x \rightarrow c}f(x) - \lim_{x \rightarrow c}g(x) (iii) \lim_{x \rightarrow c}(f(x)g(x)) = \lim_{x \rightarrow c}f(x) \times \lim_{x \rightarrow c}g(x) (iv) \lim_{x \rightarrow c} (\frac{f(x)}{g(x)}) = \frac{\lim_{x \rightarrow c}f(x)}{\lim_{x \rightarrow c}g(x)}

Problem 4: If \lim_{x \rightarrow c}(\frac{f(x)}{g(x)}) exists, then (i) both \lim_{x \rightarrow c}f(x) and \lim_{x \rightarrow a}g(x) must exist (ii) \lim_{x \rightarrow a}f(x) need not exist but \lim_{x \rightarrow a}g(x) exists. (iii) neither \lim_{x \rightarrow a}f(x) nor \lim_{x \rightarrow a}g(x) may exist (d) \lim_{x \rightarrow a}f(x) exists but \lim_{x \rightarrow a}g(x) need not exist.

Problem 5: \lim_{x \rightarrow a+}f(x)=l=\lim_{x \rightarrow a-}g(x) and \lim_{x \rightarrow a-}f(x) = m = \lim_{x \rightarrow a+}g(x) then the function (f(x)-g(x)) (i) is continuous at x=a (ii) is not continuous at x=a (iii) has a limit when x \rightarrow a but \lim_{x \rightarrow a}(f(x)-g(x))=l-m (iv) has a limit equal to l-m when x \rightarrow a

Problem 6: If \lim_{x \rightarrow a+}f(x) = l = \lim_{x \rightarrow a-}g(x) and \lim_{x \rightarrow a-}f(x)=m=\lim_{x \rightarrow a+}g(x) then the function (f(x).g(x)) (i) is continuous at x=a (ii) does not have a limit at x=a (iii) has a limit when x \rightarrow a and it is equal to l.m (iv) has a limit when x \rightarrow a but it is not equal to l.m

Problem 7: Find \lim_{x \rightarrow \frac{3.\pi}{4}}\frac{1+\tan{x}}{\cos{(2x)}}

Problem 8: Find \lim_{x \rightarrow e}\frac{\log{x}-1}{x-e}.

Problem 9: Find \lim_{x \rightarrow 0}\frac{a^{x}-b^{x}}{x}

Problem 10: Find \lim_{x \rightarrow 0}\frac{2(1-\cos{x})}{x^{2}}.

Problem 11: Find \lim_{x \rightarrow 0}\frac{\sqrt{1+x}-1}{x}

Problem 12: Find \lim_{x \rightarrow 3-}\frac{|x-3|}{x-3}

Problem 13: Find \lim_{x \rightarrow 0}(\frac{1+\tan{x}}{1+\sin{x}})^{cosec{x}}

Problem 14: Find \lim_{x \rightarrow 0} \frac{(e^{2\sqrt{x}}-1)(\tan{3\sqrt{x}})}{\sin{x}}

Problem 15: Find \lim_{x \rightarrow 0}\frac{\log{\cos{x}}}{x}

Problem 16: Find x if \lim_{x \rightarrow a}\frac{a^{x}-x^{a}}{x^{x}-a^{a}}=-1

Regards,

Nalin Pithwa

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