Question

❖ᴏɴʟʏ ᴘʀᴏᴘᴇʀ ꜱᴏʟᴠᴇᴅ ᴀɴꜱᴡᴇʀ ᴡɪᴛʜ ɢᴏᴏᴅ ᴇxᴘʟᴀɴᴀɪᴏɴ ɴᴇᴇᴅᴇᴅ
❖ ɴᴏ ꜱᴘᴀᴍᴍɪɴɢ
❖ᴏɴʟʏ ꜰᴏʀ ᴍᴏᴅᴇʀᴀᴛᴏʀꜱ, ʙʀᴀɪɴʟʏ ꜱᴛᴀʀꜱ ᴀɴᴅ ᴏᴛʜᴇʀ ʙᴇꜱᴛ ᴜꜱᴇʀꜱ​​​​​​​​​​​​​​​​​​​​​​​

Answer 1

Answer:

EVALUATE

❖ᴏɴʟʏ ᴘʀᴏᴘᴇʀ ꜱᴏʟᴠᴇᴅ ᴀɴꜱᴡᴇʀ ᴡɪᴛʜ ɢᴏᴏᴅ ᴇxᴘʟᴀɴᴀɪᴏɴ ɴᴇᴇᴅᴇᴅ

❖ ɴᴏ ꜱᴘᴀᴍᴍɪɴɢ

❖ᴏɴʟʏ ꜰᴏʀ ᴍᴏᴅᴇʀᴀᴛᴏʀꜱ, ʙʀᴀɪɴʟʏ ꜱᴛᴀʀꜱ ᴀɴᴅ ᴏᴛʜᴇʀ ʙᴇꜱᴛ ᴜꜱᴇʀꜱ

Answer 2

$\lim_{x \to \infty} \frac{2 x^{3}}{2 x^{2} + 2}$

The limit of a product/quotient is the product/quotient of limits:

$\color{red}{\lim_{x \to \infty} \frac{2 x^{3}}{2 x^{2} + 2}} = \color{red}{\lim_{x \to \infty} 2 \lim_{x \to \infty} \frac{x^{3}}{2 x^{2} + 2}}$

The limit of a constant is equal to the constant:

$\lim_{x \to \infty} \frac{x^{3}}{2 x^{2} + 2} \color{red}{\lim_{x \to \infty} 2} = \lim_{x \to \infty} \frac{x^{3}}{2 x^{2} + 2} \color{red}{\left(2\right)}$

Apply the constant multiple rule

$\bold{\lim_{x \to \infty} c f{\left(x \right)} = c \lim_{x \to \infty} f{\left(x \right)} \: with \: c=\frac{1}{2} \: and \: f{\left(x \right)} = \frac{x^{3}}{x^{2} + 1}}$

$2 \: \color{red}{\lim_{x \to \infty} \frac{x^{3}}{2 x^{2} + 2}} = 2 \color{red}{\left(\frac{\lim_{x \to \infty} \frac{x^{3}}{x^{2} + 1}}{2}\right)}$

Multiply and divide by $x^{2}$

$\color{red}{\lim_{x \to \infty} \frac{x^{3}}{x^{2} + 1}} = \color{red}{\lim_{x \to \infty} \frac{x^{2} \frac{x^{3}}{x^{2}}}{x^{2} \frac{x^{2} + 1}{x^{2}}}}$

Divide:

$\color{red}{\lim_{x \to \infty} \frac{x^{2} \frac{x^{3}}{x^{2}}}{x^{2} \frac{x^{2} + 1}{x^{2}}}} = \color{red}{\lim_{x \to \infty} \frac{x}{1 + \frac{1}{x^{2}}}}$

The limit of the quotient is the quotient of limits:

$\color{red}{\lim_{x \to \infty} \frac{x}{1 + \frac{1}{x^{2}}}} = \color{red}{\frac{\lim_{x \to \infty} x}{\lim_{x \to \infty}\left(1 + \frac{1}{x^{2}}\right)}}$

The function grows without a bound:

$\lim_{x \to \infty} x = \infty$

The limit of a sum/difference is the sum/difference of limits:

$\infty \color{red}{\lim_{x \to \infty}\left(1 + \frac{1}{x^{2}}\right)}^{-1} = \infty \color{red}{\left(\lim_{x \to \infty} 1 + \lim_{x \to \infty} \frac{1}{x^{2}}\right)}^{-1}$

The limit of a constant is equal to the constant:

$\infty \left(\lim_{x \to \infty} \frac{1}{x^{2}} + \color{red}{\lim_{x \to \infty} 1}\right)^{-1} = \infty \left(\lim_{x \to \infty} \frac{1}{x^{2}} + \color{red}{1}\right)^{-1}$

The limit of a quotient is the quotient of limits:

$\infty \left(1 + \color{red}{\lim_{x \to \infty} \frac{1}{x^{2}}}\right)^{-1} = \infty \left(1 + \color{red}{\frac{\lim_{x \to \infty} 1}{\lim_{x \to \infty} x^{2}}}\right)^{-1}$

The limit of a constant is equal to the constant:

$\infty \left(1 + \frac{\color{red}{\lim_{x \to \infty} 1}}{\lim_{x \to \infty} x^{2}}\right)^{-1} = \infty \left(1 + \frac{\color{red}{1}}{\lim_{x \to \infty} x^{2}}\right)^{-1}$

Constant divided by a very big number equals 0

$\infty \left(1 + \color{red}{1 \frac{1}{\lim_{x \to \infty} x^{2}}}\right)^{-1} = \infty \left(1 + \color{red}{\left(0\right)}\right)^{-1}$

Therefore,

$\lim_{x \to \infty} \frac{2 x^{3}}{2 x^{2} + 2} = \infty$

Answer:

$\lim_{x \to \infty} \frac{2 x^{3}}{2 x^{2} + 2}=\infty$