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Second Year Math Functions and Limits


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Class 9Class 10First YearSecond Year
2.Discussthecontinuityoff(x)atx=c:(ii)f(x)={3x1 if x<14 if x=1c=12x if x>12. Discuss the continuity of f(x) at x=c :(ii) f(x)=\left\{\begin{array}{rll}3 x-1 & \text { if } & x<1 \\ 4 & \text { if } & x=1 c=1 \\ 2 x & \text { if } & x>1\end{array}\right.

4. Express each limit in terms of e :\[\text { (x) } \operatorname{Lim}_{x \rightarrow 0} \frac{e^{1 / x}-1}{e^{1 / x}+1} x<0\]
4. Express each limit in terms of  e  :\[\text { (x) } \operatorname{Lim}_{x \rightarrow 0} \frac{e^{1 / x}-1}{e^{1 / x}+1} x<0\]

4. Express each limit in terms of e :\[\text { (x) } \operatorname{Lim}_{x \rightarrow 0} \frac{e^{1 / x}-1}{e^{1 / x}+1} x<0\]

Example 1:Graph the circle x^{2}+y^{2}=4
Example 1:Graph the circle  x^{2}+y^{2}=4

Example1:Graphthecirclex2+y2=4Example 1:Graph the circle x^{2}+y^{2}=4

4. Express each limit in terms of e :(ix) \operatorname{Lim}_{x \rightarrow \infty}\left(\frac{x}{1+x}\right)^{x}
4. Express each limit in terms of  e  :(ix)  \operatorname{Lim}_{x \rightarrow \infty}\left(\frac{x}{1+x}\right)^{x}

4.Expresseachlimitintermsofe:(ix)Limx(x1+x)x4. Express each limit in terms of e :(ix) \operatorname{Lim}_{x \rightarrow \infty}\left(\frac{x}{1+x}\right)^{x}

1. Evaluate each limit by using theorems of limits:(v) \operatorname{Lim}_{x \rightarrow 2}\left(\sqrt{x^{3}+1}-\sqrt{x^{2}+5}\right)
1. Evaluate each limit by using theorems of limits:(v)  \operatorname{Lim}_{x \rightarrow 2}\left(\sqrt{x^{3}+1}-\sqrt{x^{2}+5}\right)

1.Evaluateeachlimitbyusingtheoremsoflimits:(v)Limx2(x3+1x2+5)1. Evaluate each limit by using theorems of limits:(v) \operatorname{Lim}_{x \rightarrow 2}\left(\sqrt{x^{3}+1}-\sqrt{x^{2}+5}\right)

Example 1: Determine whether \operatorname{Lim}_{x \rightarrow 2} f(x) and \operatorname{Lim}_{x \rightarrow 4} f(x) exist when\[f(x)=\left\{\begin{array}{rcc}2 x+1 & \text { if } & 0 \leq x \leq 2 \\7-x & \text { if } & 2 \leq x \leq 4 \\x & \text { if } & 4 \leq x \leq 6\end{array}\right.\]
Example 1: Determine whether  \operatorname{Lim}_{x \rightarrow 2} f(x)  and  \operatorname{Lim}_{x \rightarrow 4} f(x)  exist when\[f(x)=\left\{\begin{array}{rcc}2 x+1 & \text { if } & 0 \leq x \leq 2 \\7-x & \text { if } & 2 \leq x \leq 4 \\x & \text { if } & 4 \leq x \leq 6\end{array}\right.\]

Example 1: Determine whether \operatorname{Lim}_{x \rightarrow 2} f(x) and \operatorname{Lim}_{x \rightarrow 4} f(x) exist when\[f(x)=\left\{\begin{array}{rcc}2 x+1 & \text { if } & 0 \leq x \leq 2 \\7-x & \text { if } & 2 \leq x \leq 4 \\x & \text { if } & 4 \leq x \leq 6\end{array}\right.\]

Example 7:Evaluate: \operatorname{Lim}_{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}
Example 7:Evaluate:  \operatorname{Lim}_{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}

Example7:Evaluate:Limθ01cosθθExample 7:Evaluate: \operatorname{Lim}_{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta}

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