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Kalkulator Limit
Hitung limit langkah demi langkah
Kalkulator gratis ini akan mencoba mencari limit (dua sisi maupun satu sisi, termasuk kiri dan kanan) dari fungsi yang diberikan pada titik yang ditentukan (termasuk tak hingga), beserta langkah-langkahnya.
Berbagai teknik digunakan untuk menangani limit (termasuk bentuk tak tentu): aturan-aturan limit, penulisan ulang dan penyederhanaan, aturan L'Hôpital, merasionalisasi penyebut, mengambil logaritma natural, dan sebagainya.
Solution
Your input: find $$$\lim_{x \to -\infty}\left(x^{3} - 3 x^{2}\right)$$$
Multiply and divide by $$$x^{3}$$$:
$${\color{red}{\lim_{x \to -\infty}\left(x^{3} - 3 x^{2}\right)}} = {\color{red}{\lim_{x \to -\infty} x^{3} \frac{x^{3} - 3 x^{2}}{x^{3}}}}$$
Divide:
$${\color{red}{\lim_{x \to -\infty} x^{3} \frac{x^{3} - 3 x^{2}}{x^{3}}}} = {\color{red}{\lim_{x \to -\infty} x^{3} \left(1 - \frac{3}{x}\right)}}$$
The limit of a product/quotient is the product/quotient of limits:
$${\color{red}{\lim_{x \to -\infty} x^{3} \left(1 - \frac{3}{x}\right)}} = {\color{red}{\lim_{x \to -\infty} x^{3} \lim_{x \to -\infty}\left(1 - \frac{3}{x}\right)}}$$
The limit of a sum/difference is the sum/difference of limits:
$$\lim_{x \to -\infty} x^{3} {\color{red}{\lim_{x \to -\infty}\left(1 - \frac{3}{x}\right)}} = \lim_{x \to -\infty} x^{3} {\color{red}{\left(\lim_{x \to -\infty} 1 - \lim_{x \to -\infty} \frac{3}{x}\right)}}$$
The limit of a constant is equal to the constant:
$$\lim_{x \to -\infty} x^{3} \left(- \lim_{x \to -\infty} \frac{3}{x} + {\color{red}{\lim_{x \to -\infty} 1}}\right) = \lim_{x \to -\infty} x^{3} \left(- \lim_{x \to -\infty} \frac{3}{x} + {\color{red}{1}}\right)$$
Apply the constant multiple rule $$$\lim_{x \to -\infty} c f{\left(x \right)} = c \lim_{x \to -\infty} f{\left(x \right)}$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:
$$\lim_{x \to -\infty} x^{3} \left(1 - {\color{red}{\lim_{x \to -\infty} \frac{3}{x}}}\right) = \lim_{x \to -\infty} x^{3} \left(1 - {\color{red}{\left(3 \lim_{x \to -\infty} \frac{1}{x}\right)}}\right)$$
The limit of a quotient is the quotient of limits:
$$\lim_{x \to -\infty} x^{3} \left(1 - 3 {\color{red}{\lim_{x \to -\infty} \frac{1}{x}}}\right) = \lim_{x \to -\infty} x^{3} \left(1 - 3 {\color{red}{\frac{\lim_{x \to -\infty} 1}{\lim_{x \to -\infty} x}}}\right)$$
The limit of a constant is equal to the constant:
$$\lim_{x \to -\infty} x^{3} \left(1 - \frac{3 {\color{red}{\lim_{x \to -\infty} 1}}}{\lim_{x \to -\infty} x}\right) = \lim_{x \to -\infty} x^{3} \left(1 - \frac{3 {\color{red}{1}}}{\lim_{x \to -\infty} x}\right)$$
Constant divided by a very big number equals $$$0$$$:
$$\lim_{x \to -\infty} x^{3} \left(1 - 3 {\color{red}{1 \frac{1}{\lim_{x \to -\infty} x}}}\right) = \lim_{x \to -\infty} x^{3} \left(1 - 3 {\color{red}{\left(0\right)}}\right)$$
The function decreases without a bound:
$$\lim_{x \to -\infty} x^{3} = -\infty$$
Therefore,
$$\lim_{x \to -\infty}\left(x^{3} - 3 x^{2}\right) = -\infty$$
Answer: $$$\lim_{x \to -\infty}\left(x^{3} - 3 x^{2}\right)=-\infty$$$