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Question: How do you use continuity to evaluate the limit \(\sin (x + \sin x)\) ?...

How do you use continuity to evaluate the limit sin⁡(x+sin⁡x)\sin (x + \sin x) ?

Explanation

Solution

We are given a function in terms of x, so as the value of x changes the value of the function also changes. In this question, we have to find the limit of the given function using continuity, that is, we have to find the value attained by the function when x takes the values very close to a given value. A function is said to be continuous when we do not have to lift our pencil/pen while graphing the function. So, we will evaluate the limit by using substitution using the knowledge about continuity.

Complete step by step solution:
Let's check the continuity of x at any random value “a”.
When lim⁡x→a−x=lim⁡h→0(a−h)=a\mathop {\lim }\limits_{x \to {a^ - }} x = \mathop {\lim }\limits_{h \to 0} (a - h) = a and lim⁡x→a+x=lim⁡h→0(a+h)=a\mathop {\lim }\limits_{x \to {a^ + }} x = \mathop {\lim }\limits_{h \to 0} (a + h) = a
As the right-hand limit is equal to the left-hand limit, so x is continuous at a, so lim⁡x→ax=a\mathop {\lim }\limits_{x \to a} x = a
We know that the sine of a continuous function is also continuous, so the sine of x is also a continuous function, that is, lim⁡x→asin⁡x=sin⁡a\mathop {\lim }\limits_{x \to a} \sin x = \sin a .
We also know that the sum of two continuous functions is also continuous.
As x and sine of x are continuous so x+sin⁡xx + \sin x is also continuous and thus the sine of x+sin⁡xx + \sin x will also be continuous.
lim⁡x→asin⁡(x+sin⁡x)=sin⁡(a+sin⁡a)\mathop {\lim }\limits_{x \to a} \sin (x + \sin x) = \sin (a + \sin a)

Hence, we can find the limit sin⁡(x+sin⁡x)\sin (x + \sin x) by using continuity as shown above.

Note: As the input value approaches some specific value from the left and the right side, the limits are called left side limit and right side limit respectively. A function is said to be continuous if the left side limit is equal to the right side limit. When a function is continuous at a point, the function’s value at that point and the limit as the variable approaches that point are always equal. Thus by proving that the given function is continuous at any point “a”, we can evaluate the limit of the given function.