## 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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