Question

Evaluate the limit \[\dfrac{{\sin 7x}}{x}\] as x approaches zero.

Answer 1

Hint: The given question deals with the concept of limits. In the given question we have to find out the limit for the given expression as x approaches zero. In order to find out the limit for the given question we will make use of properties of limits and some standard limits.

Complete step by step solution:
Let us consider the given question,
We need to find the limit \[\dfrac{{\sin 7x}}{x}\] as x approaches 0.
Let \[q = 7x\]
Therefore, we have,
\[ \Rightarrow \mathop {\lim }\limits_{q \to 0} \dfrac{{\sin q}}{{\dfrac{q}{7}}}\]
\[ \Rightarrow \mathop {\lim }\limits_{q \to 0} 7\dfrac{{\sin q}}{q}\]
\[ \Rightarrow 7\mathop {\lim }\limits_{q \to 0} \dfrac{{\sin q}}{q} - - - - - \left( 1 \right)\]
Now, we know that \[\mathop {\lim }\limits_{x \to 0} = \dfrac{{\sin x}}{x} = 1\]
Thus by the above expression we can say that, \[\mathop {\lim }\limits_{q \to 0} \dfrac{{\sin q}}{q} = 1\]

Therefore, \[ \Rightarrow 7\mathop {\lim }\limits_{q \to 0} \dfrac{{\sin q}}{q} = 7 \times 1 = 7\] Which is our required limit.

Additional information:
We were able to solve the above question with the help of trigonometric limits. Other than these trigonometric limits (explained later) there are five more very important trigonometric properties of limit. These other identities are useful in solving almost every question that involves limits. They are as follows:
Let a, k and P,Q represent real numbers and \[f\] and \[g\] be functions. Such that, \[\mathop {\lim }\limits_{x \to a} f\left( x \right)\] and \[\mathop {\lim }\limits_{x \to a} g\left( x \right)\] are limits that exist and are finite. The properties for such limits are as follows:
For a constant say, k it is: \[\mathop {\lim }\limits_{x \to a} k = k\]
Constant times a function is, \[\mathop {\lim }\limits_{x \to a} k.f\left( x \right) = k\mathop {\lim }\limits_{x \to a} f\left( x \right) = ka\]
For addition and subtraction of function, the properties are \[\mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right) + g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) + \mathop {\lim }\limits_{x \to a} g\left( x \right) = P + Q\] and \[\mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right) - g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) - \mathop {\lim }\limits_{x \to a} g\left( x \right) = P - Q\] respectively.
Last but not the least, for division and multiplication of functions, the properties are, \[\mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right).g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) \times \mathop {\lim }\limits_{x \to a} g\left( x \right)\] and \[\mathop {\lim }\limits_{x \to a} \dfrac{{f\left( x \right)}}{{g\left( x \right)}} = \dfrac{{\mathop {\lim }\limits_{x \to a} f\left( x \right)}}{{\mathop {\lim }\limits_{x \to a} g\left( x \right)}}\] respectively.

Note: As you can see we have used standard trigonometric limit i.e., \[\mathop {\lim }\limits_{x \to 0} = \dfrac{{\sin x}}{x} = 1\] . Including this there are a total four very important standard trigonometric limits. They are as follows:
1) \[\mathop {\lim }\limits_{x \to c} \sin x = \sin c\] 2) \[\mathop {\lim }\limits_{x \to c} \cos x = \cos c\] 3) \[\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin x}}{x} = 1\] 4) \[\mathop {\lim }\limits_{x \to 0} \dfrac{{1 - \cos x}}{x} = 0\]