simplify lim x ->-4 (sqrt(x^2+9)-3)/(r+4)
To simplify the given limit, let's start by substituting x = -4 into the expression:
lim x -> -4 (√(x^2+9)-3)/(x+4)
Plugging in x = -4, we have:
= (√((-4)^2+9)-3)/(-4+4)
= (√(16+9)-3)/0
Now, we have an indeterminate form of 0/0. To further simplify, we can apply L'Hôpital's rule by taking the derivative of the numerator and denominator separately:
= lim x -> -4 (d/dx(√(x^2+9)-3))/(d/dx(x+4))
= lim x -> -4 (1/2√(x^2+9) * 2x)/(1)
= lim x -> -4 (x/√(x^2+9))
Plugging in x = -4 now:
= (-4/√((-4)^2+9))
= (-4/√(16+9))
= (-4/√25)
= -4/5
Therefore, lim x -> -4 (√(x^2+9)-3)/(x+4) simplifies to -4/5.
To simplify the given limit, let's first simplify the expression inside the square root:
lim x -> -4 (sqrt(x^2 + 9) - 3) / (x + 4)
Now, substitute x = -4 into the expression:
sqrt((-4)^2 + 9) - 3 / (-4 + 4)
Simplifying further:
sqrt(16 + 9) - 3 / 0
The denominator is 0 which makes the expression undefined. Therefore, the limit does not exist in this case.
To simplify the limit `lim x -> -4 (sqrt(x^2+9)-3)/(r+4)`, we will analyze each component of the expression separately and then calculate the limit at the end.
Let's start with the numerator:
`sqrt(x^2+9)-3`
To simplify this, we need to use a common trigonometric identity: `a^2 - b^2 = (a + b)(a - b)`. In this case, we can rewrite `sqrt(x^2+9)` as `(sqrt(x^2+9) + 3)(sqrt(x^2+9) - 3)`. So, the numerator becomes `(sqrt(x^2+9) + 3)(sqrt(x^2+9) - 3) - 3`.
Now let's move on to the denominator:
`(r + 4)`
Since there is no algebraic simplification available for the denominator, it remains as it is.
Combining the numerator and the denominator, we get:
`[(sqrt(x^2+9) + 3)(sqrt(x^2+9) - 3) - 3]/(r + 4)`
Now, let's calculate the limit:
`lim x -> -4 [(sqrt(x^2+9) + 3)(sqrt(x^2+9) - 3) - 3]/(r + 4)`
To calculate the limit, we substitute `-4` for `x`:
= `[(sqrt((-4)^2+9) + 3)(sqrt((-4)^2+9) - 3) - 3]/(r + 4)`
Simplifying:
= `[(sqrt(16+9) + 3)(sqrt(16+9) - 3) - 3]/(r + 4)`
= `[(sqrt(25) + 3)(sqrt(25) - 3) - 3]/(r + 4)`
= `[(5 + 3)(5 - 3) - 3]/(r + 4)`
= `[8(2) - 3]/(r + 4)`
= `(16 - 3)/(r + 4)`
= `13/(r + 4)`
Therefore, the simplified form of the limit `lim x -> -4 (sqrt(x^2+9)-3)/(r+4)` is `13/(r + 4)`.