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Second Year Math Functions and Limits 9. Determine whether the given function f is even or odd.(iii) f(x)=x \sqrt{x^{2}+5}


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9. Determine whether the given function f is even or odd.(iii) f(x)=x \sqrt{x^{2}+5}

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

Example 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. 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. 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}

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

4. Express each limit in terms of e :(vii) \operatorname{Lim}_{x \rightarrow 0}\left(1+2 x^{2}\right)^{\frac{1}{x^{2}}}
4. Express each limit in terms of  e  :(vii)  \operatorname{Lim}_{x \rightarrow 0}\left(1+2 x^{2}\right)^{\frac{1}{x^{2}}}

4. Express each limit in terms of e :(vii) \operatorname{Lim}_{x \rightarrow 0}\left(1+2 x^{2}\right)^{\frac{1}{x^{2}}}

Example 4: Discuss the continuity of the function f(x) and g(x) at x=3 .(a) f(x)=\left\{\begin{array}{cc}\frac{x^{2}-9}{x-3} & \text { if } x \neq 3 \\ 6 & \text { if } x=3\end{array}\right.
Example 4: Discuss the continuity of the function  f(x)  and  g(x)  at  x=3 .(a)  f(x)=\left\{\begin{array}{cc}\frac{x^{2}-9}{x-3} & \text { if } x \neq 3 \\ 6 & \text { if } x=3\end{array}\right.

Example 4: Discuss the continuity of the function f(x) and g(x) at x=3 .(a) f(x)=\left\{\begin{array}{cc}\frac{x^{2}-9}{x-3} & \text { if } x \neq 3 \\ 6 & \text { if } x=3\end{array}\right.

1. Determine the left hand limit and the right hand limit and then find the limit of the following functions when x \rightarrow c (ii) f(x)=\frac{x^{2}-9}{x-3} \mathrm{c}=-3
1. Determine the left hand limit and the right hand limit and then find the limit of the following functions when  x \rightarrow c (ii)  f(x)=\frac{x^{2}-9}{x-3}  \mathrm{c}=-3

1. Determine the left hand limit and the right hand limit and then find the limit of the following functions when x \rightarrow c (ii) f(x)=\frac{x^{2}-9}{x-3} \mathrm{c}=-3

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

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

Example 2: Let f: R \rightarrow R be the function defined by\[f(x)=2 x+1 . \text { Find } f^{-1}(x)\]
Example 2: Let  f: R \rightarrow R  be the function defined by\[f(x)=2 x+1 . \text { Find } f^{-1}(x)\]

Example 2: Let f: R \rightarrow R be the function defined by\[f(x)=2 x+1 . \text { Find } f^{-1}(x)\]

2. Evaluate each limit by using algebraic techniques.(v) \operatorname{Lim}_{x \rightarrow-1}\left(\frac{x^{3}+x^{2}}{x^{2}-1}\right)
2. Evaluate each limit by using algebraic techniques.(v)  \operatorname{Lim}_{x \rightarrow-1}\left(\frac{x^{3}+x^{2}}{x^{2}-1}\right)

2. Evaluate each limit by using algebraic techniques.(v) \operatorname{Lim}_{x \rightarrow-1}\left(\frac{x^{3}+x^{2}}{x^{2}-1}\right)

1. Evaluate each limit by using theorems of limits:(iv) \operatorname{Lim}_{x \rightarrow 2} \sqrt{x^{2}-4}
1. Evaluate each limit by using theorems of limits:(iv)  \operatorname{Lim}_{x \rightarrow 2} \sqrt{x^{2}-4}

1. Evaluate each limit by using theorems of limits:(iv) \operatorname{Lim}_{x \rightarrow 2} \sqrt{x^{2}-4}

2. 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.
2. 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.

2. 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.

3. Draw the graphs of the functions defined below and find whether they are continuous.(iv) y=\frac{x^{2}-16}{x-4} x \neq 4
3. Draw the graphs of the functions defined below and find whether they are continuous.(iv)  y=\frac{x^{2}-16}{x-4} x \neq 4

3. Draw the graphs of the functions defined below and find whether they are continuous.(iv) y=\frac{x^{2}-16}{x-4} x \neq 4

2. For the real valued function f defined below find(a) f^{-1}(x) (b) f^{-1}(-1) and verify \left.f\left(f^{-1}(x)\right)=f^{-1} f(x)\right)=x (ii) f(x)=3 x^{3}+7
2. For the real valued function  f  defined below find(a)  f^{-1}(x) (b)  f^{-1}(-1)  and verify  \left.f\left(f^{-1}(x)\right)=f^{-1} f(x)\right)=x (ii)  f(x)=3 x^{3}+7

2. For the real valued function f defined below find(a) f^{-1}(x) (b) f^{-1}(-1) and verify \left.f\left(f^{-1}(x)\right)=f^{-1} f(x)\right)=x (ii) f(x)=3 x^{3}+7

Example 1:(i) \operatorname{Lim}_{x \rightarrow 1} \frac{x^{2}-1}{x^{2}-x}
Example 1:(i)  \operatorname{Lim}_{x \rightarrow 1} \frac{x^{2}-1}{x^{2}-x}

Example 1:(i) \operatorname{Lim}_{x \rightarrow 1} \frac{x^{2}-1}{x^{2}-x}

3. Express the following:(a) The perimeter P of square as a function of its area A .
3. Express the following:(a) The perimeter  P  of square as a function of its area  A .

3. Express the following:(a) The perimeter P of square as a function of its area A .

Example 5: Express each limit in terms of the number e (b) \operatorname{Lim}_{h \rightarrow 0}(1+2 h)^{\frac{1}{h}}
Example 5: Express each limit in terms of the number   e  (b)  \operatorname{Lim}_{h \rightarrow 0}(1+2 h)^{\frac{1}{h}}
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Example 5: Express each limit in terms of the number e (b) \operatorname{Lim}_{h \rightarrow 0}(1+2 h)^{\frac{1}{h}}

Example 5: Discuss continuity of f at 3when f(x)=\left\{\begin{array}{lll}x-1 & \text { if } & x<3 \\ 2 x+1 & \text { if } & 3 \leq x\end{array}\right.
Example 5:    Discuss continuity of  f  at 3when  f(x)=\left\{\begin{array}{lll}x-1 & \text { if } & x<3 \\ 2 x+1 & \text { if } & 3 \leq x\end{array}\right.
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Example 5: Discuss continuity of f at 3when f(x)=\left\{\begin{array}{lll}x-1 & \text { if } & x<3 \\ 2 x+1 & \text { if } & 3 \leq x\end{array}\right.

9. Determine whether the given function f is even or odd.(iii) f(x)=x \sqrt{x^{2}+5}
9. Determine whether the given function  f  is even or odd.(iii)  f(x)=x \sqrt{x^{2}+5}
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9. Determine whether the given function f is even or odd.(iii) f(x)=x \sqrt{x^{2}+5}

4. Express each limit in terms of e :(ii) \operatorname{Lim}_{n \rightarrow+\infty}\left(1+\frac{1}{n}\right)^{\frac{n}{2}}
4. Express each limit in terms of  e  :(ii)  \operatorname{Lim}_{n \rightarrow+\infty}\left(1+\frac{1}{n}\right)^{\frac{n}{2}}
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4. Express each limit in terms of e :(ii) \operatorname{Lim}_{n \rightarrow+\infty}\left(1+\frac{1}{n}\right)^{\frac{n}{2}}

Example3: \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a} with n \in \mathbb{N}
Example3:   \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}   with  n \in \mathbb{N}
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Example3: \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a} with n \in \mathbb{N}

3. Evaluate the following limits(iv) \operatorname{Lim}_{x \rightarrow \pi} \frac{\sin x}{\pi-x}
3. Evaluate the following limits(iv)  \operatorname{Lim}_{x \rightarrow \pi} \frac{\sin x}{\pi-x}
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3. Evaluate the following limits(iv) \operatorname{Lim}_{x \rightarrow \pi} \frac{\sin x}{\pi-x}

Example 3: Determine whether the following functions are even or odd.(c) f(x)=\sin x+\cos x
Example 3: Determine whether the following functions are even or odd.(c)  f(x)=\sin x+\cos x
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Example 3: Determine whether the following functions are even or odd.(c) f(x)=\sin x+\cos x

Example 4:(ii) \operatorname{Lim}_{x \rightarrow+\infty} \frac{2-3 x}{\sqrt{3+4 x^{2}}}
Example 4:(ii)  \operatorname{Lim}_{x \rightarrow+\infty} \frac{2-3 x}{\sqrt{3+4 x^{2}}}
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Example 4:(ii) \operatorname{Lim}_{x \rightarrow+\infty} \frac{2-3 x}{\sqrt{3+4 x^{2}}}

2. Evaluate each limit by using algebraic techniques.(iv) \operatorname{Lim}_{x \rightarrow 1} \frac{x^{3}-3 x^{2}+3 x-1}{x^{3}-x}
2. Evaluate each limit by using algebraic techniques.(iv)  \operatorname{Lim}_{x \rightarrow 1} \frac{x^{3}-3 x^{2}+3 x-1}{x^{3}-x}
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2. Evaluate each limit by using algebraic techniques.(iv) \operatorname{Lim}_{x \rightarrow 1} \frac{x^{3}-3 x^{2}+3 x-1}{x^{3}-x}

1. Evaluate each limit by using theorems of limits:(i) \operatorname{Lim}_{x \rightarrow 3}(2 x+4)
1. Evaluate each limit by using theorems of limits:(i)  \operatorname{Lim}_{x \rightarrow 3}(2 x+4)
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1. Evaluate each limit by using theorems of limits:(i) \operatorname{Lim}_{x \rightarrow 3}(2 x+4)

1. The real valued functions f and g are defined below. Find(a) f \circ g(x) (b) g \circ f(\mathrm{x}) (c) f \circ f(x) (d) g o g(\mathrm{x}) (iii) f(x)=\frac{1}{\sqrt{x-1}} x \neq 1 ; g(x)=\left(x^{2}+1\right)^{2}
1. The real valued functions  f  and  g  are defined below. Find(a)  f \circ g(x) (b)  g \circ f(\mathrm{x}) (c)  f \circ f(x) (d)  g o g(\mathrm{x}) (iii)   f(x)=\frac{1}{\sqrt{x-1}} x \neq 1  ;  g(x)=\left(x^{2}+1\right)^{2}
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1. The real valued functions f and g are defined below. Find(a) f \circ g(x) (b) g \circ f(\mathrm{x}) (c) f \circ f(x) (d) g o g(\mathrm{x}) (iii) f(x)=\frac{1}{\sqrt{x-1}} x \neq 1 ; g(x)=\left(x^{2}+1\right)^{2}

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