In this limit problem, the cosine of angle $x$ is subtracted from the sine of angle $x$ in the numerator and the angle forty five degrees is subtracted from the angle $x$ in the denominator. The limit of the quotient of them has to be calculated as the value of $x$ tends to pi by four.
Let’s try to find the limit of the $\sin{x}$ minus $\cos{x}$ by $x$ minus $\pi$ by $4$ as the value of $x$ is closer to pi by four by using the direct substitution method.
$=\,\,\,$ $\dfrac{\sin{\bigg(\dfrac{\pi}{4}\bigg)}-\cos{\bigg(\dfrac{\pi}{4}\bigg)}}{\dfrac{\pi}{4}-\dfrac{\pi}{4}}$
$=\,\,\,$ $\dfrac{\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{2}}}{\dfrac{\pi}{4}-\dfrac{\pi}{4}}$
$=\,\,\,$ $\dfrac{0}{0}$
It clears that the limit of the given rational expression is indeterminate. Hence, this limit problem should be solved in another mathematical approach.
Actually, this trigonometric limit problem can be solved in three different methods. So, let’s learn each method to know how to find the limit of the given rational function as the value of $x$ is closer to pi by four.
The expression in the numerator is a trigonometric expression, which expresses that the cosine of angle $x$ should be subtracted from the sine of angle $x$. In fact, they both are trigonometric functions, so, we cannot find the difference between them unless we know the value of angle $x$.
$\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}}$
Let’s remind all trigonometric identities. There is only one trigonometric identity in which the sine and cosine functions are involved in subtraction, and it is the sine angle difference identity but the trigonometric expression in the numerator is not exactly the same as the expansion of the sine angle difference formula. However, it is possible to adjust the trigonometric expression in the numerator into required form.
The values of both sine and cosines are equal, for the angle $45^\circ$. So, let’s try to include the both functions in the trigonometric expression of the numerator.
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(1 \times \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}\bigg)}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\dfrac{\sqrt{2}}{\sqrt{2}} \times \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}\bigg)}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\dfrac{\sqrt{2} \times 1}{\sqrt{2}} \times \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}\bigg)}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\sqrt{2} \times \dfrac{1}{\sqrt{2}} \times \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}\bigg)}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\sqrt{2} \times \dfrac{\dfrac{1}{\sqrt{2}} \times (\sin{x}-\cos{x})}{x-\dfrac{\pi}{4}}\bigg)}$
Now, the distributive property can be used for distributing one by square root of two over the subtraction of the sine and cosine functions.
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\sqrt{2} \times \dfrac{\dfrac{1}{\sqrt{2}} \times \sin{x}-\dfrac{1}{\sqrt{2}} \times \cos{x}}{x-\dfrac{\pi}{4}}\bigg)}$
According to the trigonometry, the sine of angle $45$ degrees and cosine of angle forty five degrees are equal to the multiplicative inverse of the square root of two.
$(1).\,\,\,$ $\sin{\bigg(\dfrac{\pi}{4}\bigg)} \,=\, \dfrac{1}{\sqrt{2}}$
$(2).\,\,\,$ $\cos{\bigg(\dfrac{\pi}{4}\bigg)} \,=\, \dfrac{1}{\sqrt{2}}$
Now, we can replace each reciprocal of square root of two by the above trigonometric functions.
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\sqrt{2} \times \dfrac{\cos{\bigg(\dfrac{\pi}{4}\bigg)} \times \sin{x}-\sin{\bigg(\dfrac{\pi}{4}\bigg)} \times \cos{x}}{x-\dfrac{\pi}{4}}\bigg)}$
The commutative property of multiplication can be used here for making the trigonometric expression in the numerator as the expansion of the sine of angle difference rule.
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\sqrt{2} \times \dfrac{\sin{x} \times \cos{\bigg(\dfrac{\pi}{4}\bigg)}- \cos{x} \times \sin{\bigg(\dfrac{\pi}{4}\bigg)}}{x-\dfrac{\pi}{4}}\bigg)}$
Finally, the trigonometric expression in the numerator expresses the expansion of the sine of angle difference identity.
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \bigg(\sqrt{2} \times \dfrac{\sin{\bigg(x-\dfrac{\pi}{4}\bigg)}}{x-\dfrac{\pi}{4}}\bigg)}$
In calculus, we have a trigonometric limit rule in terms of sine function. So, let’s try to adjust the simplified function into the form of the trigonometric limit rule. Firstly, let’s separate the multiplicative constant from the function as per the constant multiple rule of limits.
$=\,\,\,$ $\sqrt{2} \times \displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\sin{\bigg(x-\dfrac{\pi}{4}\bigg)}}{x-\dfrac{\pi}{4}}}$
If $x\,\to\, \dfrac{\pi}{4}$, then $x-\dfrac{\pi}{4}\,\to\, \dfrac{\pi}{4}-\dfrac{\pi}{4}$, Therefore $x-\dfrac{\pi}{4}\,\to\, 0$, which clears that when $x$ approaches to $\dfrac{\pi}{4}$, the value of $x-\dfrac{\pi}{4}$ is closer to zero.
$=\,\,\,$ $\sqrt{2} \times \displaystyle \large \lim_{\Large x\,-\,\normalsize \dfrac{\pi}{4} \Large \,\to\, 0}{\normalsize \dfrac{\sin{\bigg(x-\dfrac{\pi}{4}\bigg)}}{x-\dfrac{\pi}{4}}}$
For our convenience, let us take $y \,=\, x-\dfrac{\pi}{4}$ and express the whole function in terms of $y$.
$=\,\,\,$ $\sqrt{2} \times \displaystyle \large \lim_{\Large y \,\to\, 0}{\normalsize \dfrac{\sin{y}}{y}}$
According to the trigonometric limit rule in sine function, the quotient of sine of angle $y$ by $y$ as the value of $y$ is closer to zero is equal to one.
$=\,\,\,$ $\sqrt{2} \times 1$
$=\,\,\,$ $\sqrt{2}$
We can also evaluate the limit of the quotient of the subtraction $\cos{x}$ from $\sin{x}$ by the subtraction of pi by four from the value $x$ as the variable $x$ approaches to pi by four by the combination of both trigonometric identities and a trigonometric limit rule.
The trigonometric functions $\sin{x}$ and $\cos{x}$ cannot be expanded. So, if we make some setting to expand each function, then we can easily simplify the trigonometric expression in the numerator of the given rational expression and it helps us to find the limit of that function easily.
$\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}}$
Let us take $y \,=\, x-\dfrac{\pi}{4}$, then $x \,=\, \dfrac{\pi}{4}+y$. Now, we can write the given function in terms of $y$.
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\normalsize \,+\, \dfrac{\pi}{4}\Large \,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\sin{\bigg(\dfrac{\pi}{4}+y\bigg)}-\cos{\bigg(\dfrac{\pi}{4}+y\bigg)}}{y}}$
If $y+\dfrac{\pi}{4} \,\to\, \dfrac{\pi}{4}$, then $y+\dfrac{\pi}{4}-\dfrac{\pi}{4} \,\to\, \dfrac{\pi}{4}-\dfrac{\pi}{4}$. Theerfore, $y \,\to\, 0$. It means, when $y+\dfrac{\pi}{4}$ approaches to $\dfrac{\pi}{4}$, the value of $y$ tends to zero.
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\sin{\bigg(\dfrac{\pi}{4}+y\bigg)}-\cos{\bigg(\dfrac{\pi}{4}+y\bigg)}}{y}}$
The trigonometric functions sine and cosine are appearing with compound angles in the numerator. So, they can be expanded by the sin angle sum identity and cos angle sum formula.
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\sin{\bigg(\dfrac{\pi}{4}\bigg)}\cos{y}+\cos{\bigg(\dfrac{\pi}{4}\bigg)}\sin{y}-\Bigg(\cos{\bigg(\dfrac{\pi}{4}\bigg)}\cos{y}-\sin{\bigg(\dfrac{\pi}{4}\bigg)}\sin{y}\Bigg)}{y}}$
The trigonometric expression in the numerator has to be simplified to find the limit of the function.
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\sin{\bigg(\dfrac{\pi}{4}\bigg)}\cos{y}+\cos{\bigg(\dfrac{\pi}{4}\bigg)}\sin{y}-\cos{\bigg(\dfrac{\pi}{4}\bigg)}\cos{y}+\sin{\bigg(\dfrac{\pi}{4}\bigg)}\sin{y}}{y}}$
Now, substitute the values of sine of pi by four and cosine of pi by four in the expression for simplifying the trigonometric expression.
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\dfrac{1}{\sqrt{2}} \times \cos{y}+\dfrac{1}{\sqrt{2}} \times \sin{y}-\dfrac{1}{\sqrt{2}} \times \cos{y}+\dfrac{1}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\dfrac{1}{\sqrt{2}} \times \cos{y}-\dfrac{1}{\sqrt{2}} \times \cos{y}+\dfrac{1}{\sqrt{2}} \times \sin{y}+\dfrac{1}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\cancel{\dfrac{1}{\sqrt{2}} \times \cos{y}}-\cancel{\dfrac{1}{\sqrt{2}} \times \cos{y}}+2 \times \dfrac{1}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{2 \times \dfrac{1}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\dfrac{2 \times 1}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\dfrac{2}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\dfrac{(\sqrt{2})^2}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\dfrac{\sqrt{2} \times \sqrt{2}}{\sqrt{2}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\dfrac{\cancel{\sqrt{2}} \times \sqrt{2}}{\cancel{\sqrt{2}}} \times \sin{y}}{y}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\sqrt{2} \times \sin{y}}{y}}$
The simplification process is completed successfully and it is time to find the limit of the simplified function.
$=\,\,\,$ $\displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \bigg(\sqrt{2} \times \dfrac{\sin{y}}{y}\bigg)}$
Before evaluating the limit, we have to separate the constants from the function and it can be done by the constant multiple rule of the limits.
$=\,\,\,$ $\sqrt{2} \times \displaystyle \large \lim_{\Large y\,\to\, 0}{\normalsize \dfrac{\sin{y}}{y}}$
We can evaluate the limit of the rational function by the trigonometric limit rule in sine function.
$=\,\,\,$ $\sqrt{2} \times 1$
$=\,\,\,$ $\sqrt{2}$
The limit of the given trigonometric rational expression is indeterminate as the value of variable $x$ tends to $\pi$ by $4$. Hence, L’Hospital’s rule can be used to find the limit.
According to the L’Hopital’s rule, the limit of the rational expression can be evaluated by finding the limit of the rational expression after differentiating the trigonometric expression $\sin{x}-\cos{x}$ in the numerator and the algebraic expression in the denominator.
$\implies$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\sin{x}-\cos{x}}{x-\dfrac{\pi}{4}}}$ $\,=\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\dfrac{d}{dx}{\big(\sin{x}-\cos{x}\big)}}{\dfrac{d}{dx}{\bigg(x-\dfrac{\pi}{4}\bigg)}}}$
The derivative can be distributed to each term of each expression in the mathematical expression in ratio form by the distributive property of multiplication over subtraction.
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\dfrac{d}{dx}{\big(\sin{x}\big)}-\dfrac{d}{dx}{\big(\cos{x}\big)}}{\dfrac{d}{dx}{\big(x\big)}-\dfrac{d}{dx}{\bigg(\dfrac{\pi}{4}\bigg)}}}$
The differentiation can be performed by the derivative rules of sine, cosine, variable and a constant.
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\cos{x}-\big(-\sin{x}\big)}{1-0}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \dfrac{\cos{x}+\sin{x}}{1}}$
$=\,\,\,$ $\displaystyle \large \lim_{\Large x\,\to\, \normalsize \dfrac{\pi}{4}}{\normalsize \big(\cos{x}+\sin{x}\big)}$
The given rational expression becomes a trigonometric expression by the differentiation and its limit can be evaluated by the direct substitution method as the value of $x$ approaches to pi by $4$.
$=\,\,\,$ $\cos{\bigg(\dfrac{\pi}{4}\bigg)}+\sin{\bigg(\dfrac{\pi}{4}\bigg)}$
Now, we can substitute the values of cosine of angle $45$ degrees and sine of angle forty five degrees.
$=\,\,\,$ $\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}$
Finally, simplify the mathematical expression to find the limit of the given rational function.
$=\,\,\,$ $2 \times \dfrac{1}{\sqrt{2}}$
$=\,\,\,$ $\dfrac{2 \times 1}{\sqrt{2}}$
$=\,\,\,$ $\dfrac{2}{\sqrt{2}}$
$=\,\,\,$ $\dfrac{\big(\sqrt{2}\big)^2}{\sqrt{2}}$
$=\,\,\,$ $\dfrac{\cancel{\big(\sqrt{2}\big)^2}}{\cancel{\sqrt{2}}}$
$=\,\,\, \sqrt{2}$
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