Advertisements
Advertisements
Question
Evaluate the following limit:
\[\lim_{x \to 0} \frac{\sin\left( \alpha + \beta \right)x + \sin\left( \alpha - \beta \right)x + \sin2\alpha x}{\cos^2 \beta x - \cos^2 \alpha x}\]
Advertisements
Solution
\[\lim_{x \to 0} \frac{\sin\left( \alpha + \beta \right)x + \sin\left( \alpha - \beta \right)x + \sin2\alpha x}{\cos^2 \beta x - \cos^2 \alpha x}\]
`therefore cos^2 beta x - cos^2alpha x`
`= cos^2betax - cos^2alphax + cos^2betax * cos^2alphax - cos^2betax * cos^2alphax`
= `cos^2 beta x - cos^2 beta x * cos^2 alpha x - cos^2 alpha x + cos^2 beta x * cos^2 alpha x`
= `cos^2 beta x (1 - cos^2 alpha x) - cos^2 alpha x (1 - cos^2 beta x)`
= `cos^2 beta x * sin^2 alpha x - cos^2 alpha x * sin^2 beta x`
= `(cos beta x * sin alpha x)^2 - (cos alpha x * sin beta x)`
= `(cos beta x * sin alpha x + cos alpha x * sin beta x)(cos beta x * sin alpha x - cos alpha x * sin beta x)`
= `sin (alpha x + beta x) sin (alphax - betax)` ...[sin (A ± B) = sin A cos B ± cos A sin B]
= `sin (alpha + beta)x sin (alpha - beta)x`
`therefore lim_(x->0) (x{sin (alpha + beta )x + sin (alpha - beta)x + sin 2 alpha x})/(sin (alpha + beta)x sin (alpha - beta) x)`
`lim_(x->0) ((x{sin (alpha + beta )x + sin (alpha - beta)x + sin 2 alpha x})/x^2)/((sin (alpha + beta)x sin (alpha - beta) x)/x^2)`
`lim_(x->0) ((sin (alpha + beta)x)/x + (sin (alpha - beta)x)/x + (sin 2 alpha x)/x)/((sin (alpha + beta)x)/x xx (sin (alpha - beta)x)/x)`
`lim_(x->0) ((sin (alpha + beta)x)/((alpha + beta) x) xx (alpha + beta) + (sin (alpha - beta)x)/((alpha - beta)x) xx (alpha - beta) + (sin 2 alpha x)/(2alphax) xx 2alpha)/((sin (alpha + beta)x)/((alpha + beta)x) xx (alpha + beta) xx (sin (alpha - beta)x)/((alpha - beta)x) xx (alpha - beta))`
`= (alpha + beta + alpha - beta + 2alpha)/((alpha + beta)(alpha - beta))`
`= (4alpha)/(alpha^2 - beta^2)`
APPEARS IN
RELATED QUESTIONS
\[\lim_{x \to - 1} \frac{x^3 - 3x + 1}{x - 1}\]
\[\lim_{x \to 0} \frac{ax + b}{cx + d}, d \neq 0\]
\[\lim_{x \to 2} \frac{x^4 - 16}{x - 2}\]
\[\lim_{x \to \sqrt{2}} \frac{x^2 - 2}{x^2 + \sqrt{2}x - 4}\]
\[\lim_{x \to \sqrt{3}} \frac{x^2 - 3}{x^2 + 3 \sqrt{3}x - 12}\]
\[\lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{4}{x^3 - 2 x^2} \right)\]
\[\lim_{x \to a} \frac{x^{2/7} - a^{2/7}}{x - a}\]
If \[\lim_{x \to 3} \frac{x^n - 3^n}{x - 3} = 108,\] find the value of n.
\[\lim_{x \to \infty} \frac{\left( 3x - 1 \right) \left( 4x - 2 \right)}{\left( x + 8 \right) \left( x - 1 \right)}\]
\[\lim_{x \to \infty} \frac{3 x^3 - 4 x^2 + 6x - 1}{2 x^3 + x^2 - 5x + 7}\]
\[f\left( x \right) = \frac{a x^2 + b}{x^2 + 1}, \lim_{x \to 0} f\left( x \right) = 1\] and \[\lim_{x \to \infty} f\left( x \right) = 1,\]then prove that f(−2) = f(2) = 1
\[\lim_{x \to 0} \frac{x^2}{\sin x^2}\]
\[\lim_{x \to 0} \frac{\tan mx}{\tan nx}\]
\[\lim_{x \to 0} \frac{1 - \cos 2x + \tan^2 x}{x \sin x}\]
\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos x}\]
\[\lim_{x \to 0} \frac{x^2 + 1 - \cos x}{x \sin x}\]
\[\lim_\theta \to 0 \frac{1 - \cos 4\theta}{1 - \cos 6\theta}\]
\[\lim_{x \to 0} \frac{\tan 2x - \sin 2x}{x^3}\]
\[\lim_{x \to \pi} \frac{\sin x}{\pi - x}\]
\[\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{x - \frac{\pi}{4}}\]
\[\lim_{x \to \frac{\pi}{3}} \frac{\sqrt{3} - \tan x}{\pi - 3x}\]
\[\lim_{x \to 1} \frac{1 - x^2}{\sin 2\pi x}\]
\[\lim_{n \to \infty} 2^{n - 1} \sin \left( \frac{a}{2^n} \right)\]
\[\lim_{x \to - 1} \frac{x^2 - x - 2}{\left( x^2 + x \right) + \sin \left( x + 1 \right)}\]
\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} - \cos x - \sin x}{\left( 4x - \pi \right)^2}\]
\[\lim_{x \to \frac{\pi}{4}} \frac{{cosec}^2 x - 2}{\cot x - 1}\]
\[\lim_{x \to 0} \frac{8^x - 2^x}{x}\]
Write the value of \[\lim_{x \to 0^-} \left[ x \right] .\]
\[\lim_{x \to 0^-} \frac{\sin x}{\sqrt{x}} .\]
\[\lim_{x \to } \frac{1 - \cos 2x}{x} is\]
\[\lim_{x \to \infty} \frac{\sin x}{x}\] equals
\[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}\] is equal to
The value of \[\lim_{x \to \pi/2} \left( \sec x - \tan x \right)\]is
The value of \[\lim_{x \to \infty} \frac{\left( x + 1 \right)^{10} + \left( x + 2 \right)^{10} + . . . + \left( x + 100 \right)^{10}}{x^{10} + {10}^{10}}\] is
If \[f\left( x \right) = \begin{cases}\frac{\sin\left[ x \right]}{\left[ x \right]}, & \left[ x \right] \neq 0 \\ 0, & \left[ x \right] = 0\end{cases}\] where denotes the greatest integer function, then \[\lim_{x \to 0} f\left( x \right)\]
Evaluate the following Limit:
`lim_(x -> 0) ((1 + x)^"n" - 1)/x`
If `lim_(x -> 1) (x^4 - 1)/(x - 1) = lim_(x -> k) (x^3 - l^3)/(x^2 - k^2)`, then find the value of k.
If f(x) = `{{:(1 if x "is rational"),(-1 if x "is rational"):}` is continuous on ______.
Evaluate the following limit:
`lim_(x->7)[((root(3)(x)-root(3)(7))(root(3)(x)+root(3)(7)))/(x-7)]`
