Differentiability and Continuity of Function
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Limits, Continuity and Differentiability

79922 If \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}+\frac{b}{x^{2}}\right)^{2 x}=e^{2}\), then

1 \(a=2, b=1\)
2 \(a=1, b=2\)
3 \(a=1, b \in R\)
4 \(a=b=1\)
COMEDK-2018
Limits, Continuity and Differentiability

79923 If \(\lim _{x \rightarrow \frac{1}{2}} \frac{32 x^{5}-1}{2 x-1}=\lim _{\theta \rightarrow 0} \frac{\tan (\theta / m)}{\theta}\), then \(m\) is equal to

1 2
2 5
3 \(\frac{1}{5}\)
4 \(\frac{1}{2}\)
COMEDK-2019
Limits, Continuity and Differentiability

79924 If \(f(x)=\left\{\begin{array}{cc}\frac{e^{e / x}-e^{-e / x}}{e^{1 / x}+e^{-1 / x}}, x \neq 1 \\ k, x=1\end{array}\right.\) then

1 \(f\) is continuous at \(x\), when \(k=0\)
2 \(f\) is not continuous at \(x=0\) for any real \(k\).
3 \(\lim _{x \rightarrow 0} \mathrm{f}(\mathrm{x})\) exist infinitely
4 None of these
BITSAT-2019
Limits, Continuity and Differentiability

79925 Let \(f\) be the function defined by
\(f(x)=\left\{\begin{array}{l}\frac{x^{2}-1}{x^{2}-2|x-1|-1}, x \neq 1 \\ 1 / 2, \quad x=1\end{array}\right.\)

1 The function is continuous for all values of \(x\)
2 The function is continuous only for \(x>1\)
3 The function is continuous at \(x=1\)
4 The function is not continuous at \(x=1\)
VITEEE-2019
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Limits, Continuity and Differentiability

79922 If \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}+\frac{b}{x^{2}}\right)^{2 x}=e^{2}\), then

1 \(a=2, b=1\)
2 \(a=1, b=2\)
3 \(a=1, b \in R\)
4 \(a=b=1\)
COMEDK-2018
Limits, Continuity and Differentiability

79923 If \(\lim _{x \rightarrow \frac{1}{2}} \frac{32 x^{5}-1}{2 x-1}=\lim _{\theta \rightarrow 0} \frac{\tan (\theta / m)}{\theta}\), then \(m\) is equal to

1 2
2 5
3 \(\frac{1}{5}\)
4 \(\frac{1}{2}\)
COMEDK-2019
Limits, Continuity and Differentiability

79924 If \(f(x)=\left\{\begin{array}{cc}\frac{e^{e / x}-e^{-e / x}}{e^{1 / x}+e^{-1 / x}}, x \neq 1 \\ k, x=1\end{array}\right.\) then

1 \(f\) is continuous at \(x\), when \(k=0\)
2 \(f\) is not continuous at \(x=0\) for any real \(k\).
3 \(\lim _{x \rightarrow 0} \mathrm{f}(\mathrm{x})\) exist infinitely
4 None of these
BITSAT-2019
Limits, Continuity and Differentiability

79925 Let \(f\) be the function defined by
\(f(x)=\left\{\begin{array}{l}\frac{x^{2}-1}{x^{2}-2|x-1|-1}, x \neq 1 \\ 1 / 2, \quad x=1\end{array}\right.\)

1 The function is continuous for all values of \(x\)
2 The function is continuous only for \(x>1\)
3 The function is continuous at \(x=1\)
4 The function is not continuous at \(x=1\)
VITEEE-2019
NEET DIGITAL LIBRARY
Limits, Continuity and Differentiability

79922 If \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}+\frac{b}{x^{2}}\right)^{2 x}=e^{2}\), then

1 \(a=2, b=1\)
2 \(a=1, b=2\)
3 \(a=1, b \in R\)
4 \(a=b=1\)
COMEDK-2018
Limits, Continuity and Differentiability

79923 If \(\lim _{x \rightarrow \frac{1}{2}} \frac{32 x^{5}-1}{2 x-1}=\lim _{\theta \rightarrow 0} \frac{\tan (\theta / m)}{\theta}\), then \(m\) is equal to

1 2
2 5
3 \(\frac{1}{5}\)
4 \(\frac{1}{2}\)
COMEDK-2019
Limits, Continuity and Differentiability

79924 If \(f(x)=\left\{\begin{array}{cc}\frac{e^{e / x}-e^{-e / x}}{e^{1 / x}+e^{-1 / x}}, x \neq 1 \\ k, x=1\end{array}\right.\) then

1 \(f\) is continuous at \(x\), when \(k=0\)
2 \(f\) is not continuous at \(x=0\) for any real \(k\).
3 \(\lim _{x \rightarrow 0} \mathrm{f}(\mathrm{x})\) exist infinitely
4 None of these
BITSAT-2019
Limits, Continuity and Differentiability

79925 Let \(f\) be the function defined by
\(f(x)=\left\{\begin{array}{l}\frac{x^{2}-1}{x^{2}-2|x-1|-1}, x \neq 1 \\ 1 / 2, \quad x=1\end{array}\right.\)

1 The function is continuous for all values of \(x\)
2 The function is continuous only for \(x>1\)
3 The function is continuous at \(x=1\)
4 The function is not continuous at \(x=1\)
VITEEE-2019
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Limits, Continuity and Differentiability

79922 If \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}+\frac{b}{x^{2}}\right)^{2 x}=e^{2}\), then

1 \(a=2, b=1\)
2 \(a=1, b=2\)
3 \(a=1, b \in R\)
4 \(a=b=1\)
COMEDK-2018
Limits, Continuity and Differentiability

79923 If \(\lim _{x \rightarrow \frac{1}{2}} \frac{32 x^{5}-1}{2 x-1}=\lim _{\theta \rightarrow 0} \frac{\tan (\theta / m)}{\theta}\), then \(m\) is equal to

1 2
2 5
3 \(\frac{1}{5}\)
4 \(\frac{1}{2}\)
COMEDK-2019
Limits, Continuity and Differentiability

79924 If \(f(x)=\left\{\begin{array}{cc}\frac{e^{e / x}-e^{-e / x}}{e^{1 / x}+e^{-1 / x}}, x \neq 1 \\ k, x=1\end{array}\right.\) then

1 \(f\) is continuous at \(x\), when \(k=0\)
2 \(f\) is not continuous at \(x=0\) for any real \(k\).
3 \(\lim _{x \rightarrow 0} \mathrm{f}(\mathrm{x})\) exist infinitely
4 None of these
BITSAT-2019
Limits, Continuity and Differentiability

79925 Let \(f\) be the function defined by
\(f(x)=\left\{\begin{array}{l}\frac{x^{2}-1}{x^{2}-2|x-1|-1}, x \neq 1 \\ 1 / 2, \quad x=1\end{array}\right.\)

1 The function is continuous for all values of \(x\)
2 The function is continuous only for \(x>1\)
3 The function is continuous at \(x=1\)
4 The function is not continuous at \(x=1\)
VITEEE-2019
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