lim x---->-5 x+1/x+5

Question

not sure....................

Answer 1

A 2 sided limit does not exist as the limit from the right is + infinity and the limit from the left is - infinity

Answer 2

If x tends to -5 then the numerator tends to -5+1=-4 so it tends to a negative quantity.

if you consider the denominator you have two possibilities

1) Assume that you are approaching x=-5 but for values which are greater than x=-5.

For instance, you consider x=-4.5, x=-4.6, x=-4.8, etc etc....in doing this you are going towards x=-5 but for values greater than x=-5. In this case the quantity x+5 tends to zero but always remaining positive.

For instance if x=-4.9, then x+5=0.1, if x=-4.99, then x+5=0.01 and so on. Therefore, the limit will be the ratio between a negative quantity and a positive quantity (which tends to zero), and the overall limit tends to negative infinity

2) Assume that you are approaching x=-5 but for values which are smaller than x=-5.

For instance, you consider x=-5.5, x=-5.3, x=-5.1, etc etc....in doing this you are going towards x=-5 but for values smaller than x=-5. In this case the quantity x+5 tends to zero but always remaining negative.

For instance if x=-5.1, then x+5=-0.1, if x=-5.01, then x+5=-0.01 and so on. Therefore, the limit will be the ratio between a negative quantity and another negative quantity (which tends to zero), and the overall limit tends to plus infinity

To conclude the limit of the function at x=-5 is infinity. The sign depends on the direction you're approaching x=-5

Best,

Davide

Answer 3

lim (x+1)/(x+5)

x→-5

The best way to approach this is to plot this or take points on either side of x = -5. Looks like a hyperbola with an asymptote at x=-5

If you take the limit from the right, the limit approaches -∞

If you take the limit from the left, the limit approaches +∞

Answer 4

This type (f(x)/g(x)) of limits is generally solved by graphing and enforcing the definition of limit.

That is to say:

lim_x->-5^-(x+1/x+5) = lim_x->-5⁺(x+1/x+5) <=====> lim_x->-5(x+1/x+5)

(+∞) ≠ ( -∞) No existe