Question

How do you find the limit of \[\dfrac{{\sin {{\left( t \right)}^2}}}{t}\] as \[t\] approaches to 0?

Answer 1

Hint: Apply L'Hospital’s rule to evaluate the equation. Let us know the statement of L'Hospital’s rule: It states that the limit when we divide one function by another is the same after we take the derivative of each function and to find the limit of the given function \[\dfrac{{\sin {{\left( t \right)}^2}}}{t}\], apply L’Hospital’s rule.
It is denoted as:
\[\mathop {\lim }\limits_{x \to c} \dfrac{{f(x)}}{{g(x)}} = \mathop {\lim }\limits_{x \to c} \dfrac{{f'(x)}}{{g'(x)}}\]

Complete step by step answer:
Let us write the given function as
\[\dfrac{{\sin {{\left( t \right)}^2}}}{t}\]
Considering the numerator term as
\[\sin {\left( t \right)^2} = 0\] at t = 0.
Applying L’Hospital’s rule we get
\[\mathop {\lim }\limits_{t \to 0} \dfrac{{\sin {{\left( t \right)}^2}}}{t} = \mathop {\lim }\limits_{t \to 0} \dfrac{{\cos {{\left( t \right)}^2} \cdot 2t}}{1}\]
Simplifying we get
\[\mathop {\lim }\limits_{t \to 0} \dfrac{{\sin {{\left( t \right)}^2}}}{t} = \dfrac{{1 \cdot 0}}{1}\]

Therefore, the limit of \[\dfrac{{\sin {{\left( t \right)}^2}}}{t}\] as \[t\] approaches to 0 is
\[\mathop {\lim }\limits_{t \to 0} \dfrac{{\sin {{\left( t \right)}^2}}}{t} = 0\]

Additional information:
Here are some of the properties to find the limit functions:
Sum Rule: This rule states that the limit of the sum of two functions is equal to the sum of their limits.
Constant Function Rule: The limit of a constant function is the constant.
Constant Multiple Rule: The limit of a constant times a function is equal to the product of the constant and the limit of the function.
Product Rule: This rule says that the limit of the product of two functions is the product of their limits (if they exist).
Quotient Rule: The limit of quotient of two functions is the quotient of their limits, provided that the limit in the denominator function is not zero.

Note: For a limit approaching the given value, the original functions must be differentiable either side of value, but not necessarily at the value given. The limit of a quotient is equal to the quotient of the limits. The limit of a constant function is equal to the constant. The limit of a linear function is equal to the number x is approaching.