Advertisements
Advertisements
प्रश्न
`lim_(y -> 0) ((x + y) sec(x + y) - x sec x)/y`
उत्तर
`lim_(y -> 0) ((x + y) sec(x + y) - x sec x)/y`
= `lim_(y -> 0) (x sec(x + y) + y sec (x + y) - x sec x)/y`
= `lim_(y -> 0) ([x sec (x + y) - x sec x])/y + lim_(y -> 0) (y sec (x + y))/y`
= `lim_(y -> 0) (x[sec(x + y) - sec x])/y + lim_(y -> 0) sec (x + y)`
= `lim_(y -> 0) (x[1/(cos(x + y)) - 1/cosx])/y + lim_(y -> 0) sec(x + y)`
= `lim_(y -> 0) x[(cosx - cos(x + y))/(y * cos(x + y) * cos x)] + lim_(y -> 0) sec(x + y)`
= `lim_(y -> 0) (x[-2 sin ((x + x + y)/2) * sin ((x - x - y)/2)])/(y cos(x + y) * cos x) + lim_(y -> 0) sec(x + y)`
= `(x[- 2 sin (x + y/2) * sin(- y/2)])/(cos(x + y) * cos x * y) + lim_(y -> 0) sec(x + y)`
= `lim_((y -> 0),(because y/2 -> 0)) x[([2 sin (x + y/2) sin (y/2)])/(cos (x + y) * cos x * (y/2) * 2)] + lim_(y -> 0) sec(x + y)`
∴ Taking the limits we have
= `x[sin x * 1/(cosx * cos x)] + sec x`
= `x sec x tan x + sec x`
= `sec x(x tan x + 1)`
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit.
`lim_(x ->0) cos x/(pi - x)`
Evaluate the following limit.
`lim_(x -> 0) (cos 2x -1)/(cos x - 1)`
Evaluate the following limit :
`lim_(x -> pi/6) [(2 - "cosec"x)/(cot^2x - 3)]`
Evaluate the following limit :
`lim_(x -> pi/6) [(2sin^2x + sinx - 1)/(2sin^2x - 3sinx + 1)]`
Evaluate the following :
`lim_(x -> 0)[(secx^2 - 1)/x^4]`
Evaluate the following :
`lim_(x -> "a") [(x cos "a" - "a" cos x)/(x - "a")]`
`lim_{x→-5} (sin^-1(x + 5))/(x^2 + 5x)` is equal to ______
Evaluate `lim_(x -> a) (sqrt(a + 2x) - sqrt(3x))/(sqrt(3a + x) - 2sqrt(x))`
Evaluate `lim_(x -> 0) (cos ax - cos bx)/(cos cx - 1)`
Find the derivative of f(x) = `sqrt(sinx)`, by first principle.
If f(x) = x sinx, then f" `pi/2` is equal to ______.
Evaluate: `lim_(x -> 1/2) (4x^2 - 1)/(2x - 1)`
Evaluate: `lim_(x -> 0) ((x + 2)^(1/3) - 2^(1/3))/x`
Evaluate: `lim_(x -> 1) (x^4 - sqrt(x))/(sqrt(x) - 1)`
Evaluate: `lim_(x -> 0) (sin^2 2x)/(sin^2 4x)`
Evaluate: `lim_(x -> 0) (2 sin x - sin 2x)/x^3`
`x^(2/3)`
`lim_(x -> 0) ((sin(alpha + beta) x + sin(alpha - beta)x + sin 2alpha x))/(cos 2betax - cos 2alphax) * x`
`lim_(x -> pi) sinx/(x - pi)` is equal to ______.
`lim_(x -> 1) (x^m - 1)/(x^n - 1)` is ______.
`lim_(x -> 0) ("cosec" x - cot x)/x` is equal to ______.
If `f(x) = {{:(sin[x]/[x]",", [x] ≠ 0),(0",", [x] = 0):}`, where [.] denotes the greatest integer function, then `lim_(x -> 0) f(x)` is equal to ______.
If `f(x) = {{:(x^2 - 1",", 0 < x < 2),(2x + 3",", 2 ≤ x < 3):}`, the quadratic equation whose roots are `lim_(x -> 2^-) f(x)` and `lim_(x -> 2^+) f(x)` is ______.
The value of `lim_(x → ∞) ((x^2 - 1)sin^2(πx))/(x^4 - 2x^3 + 2x - 1)` is equal to ______.
Let Sk = `sum_(r = 1)^k tan^-1(6^r/(2^(2r + 1) + 3^(2r + 1)))`. Then `lim_(k→∞)` Sk = is equal to ______.
`lim_(x rightarrow ∞) sum_(x = 1)^20 cos^(2n) (x - 10)` is equal to ______.
`lim_(x rightarrow π/2) ([1 - tan (x/2)] (1 - sin x))/([1 + tan (x/2)] (π - 2x)^3` is ______.
