Calculadora de límites
Calcular límites paso a paso
Esta calculadora gratuita intentará hallar el límite (bilateral o unilateral, por la izquierda o por la derecha) de la función dada en el punto indicado (incluido el infinito), mostrando los pasos.
Se emplean distintas técnicas para tratar límites (incluidas las formas indeterminadas): leyes de los límites, reescribir y simplificar, regla de L'Hôpital, racionalizar el denominador, tomar el logaritmo natural, etc.
Solution
Your input: find $$$\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x}$$$
Rewrite:
$${\color{red}{\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x}}} = {\color{red}{\lim_{x \to \infty}\left(1 + \frac{\sin{\left(x \right)}}{x}\right)}}$$
The limit of a sum/difference is the sum/difference of limits:
$${\color{red}{\lim_{x \to \infty}\left(1 + \frac{\sin{\left(x \right)}}{x}\right)}} = {\color{red}{\left(\lim_{x \to \infty} 1 + \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x}\right)}}$$
The limit of a constant is equal to the constant:
$$\lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} + {\color{red}{\lim_{x \to \infty} 1}} = \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} + {\color{red}{1}}$$
Since the absolute value of the sine is is not greater than $$$1$$$, then:
$$- \frac{1}{x} \leq \frac{\sin{\left(x \right)}}{x} \leq \frac{1}{x}$$
Taking the limits, we have that:
$$\lim_{x \to \infty}\left(- \frac{1}{x}\right) \leq \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} \leq \lim_{x \to \infty} \frac{1}{x}$$
$$0 \leq \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} \leq 0$$
Since the limits are equal, then, by the Squeeze Theorem:
$$\lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x}=0$$
Therefore,
$$\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x} = 1$$
Answer: $$$\lim_{x \to \infty} \frac{x + \sin{\left(x \right)}}{x}=1$$$