find lim y. and lim y
x->infinity. x-> negative infinity
1.) y= (x + sinx)/x
2.) y= x/ (1+ 2^x)
1) y= x/x + sinx/x=1+sinx /x
2) 2^x>>x
for instance, x=5
2^x=32,>5
2^56=7.2E15>>56
Lim x>>inf= x/2^x >0
lim x>>-inf= x/1= -inf
thank you :)
To find the limit of y as x approaches infinity and negative infinity in each given function, we can either use algebraic simplification or apply certain limit rules. Let's calculate them one by one:
1.) y = (x + sinx)/x
To find the limit as x approaches infinity, we can divide every term by x, which won't change the limit:
y = (x/x) + (sinx/x)
y = 1 + (sinx/x)
Now, check the limit of each term separately:
The limit of 1 as x approaches infinity is simply 1.
To find the limit of (sinx/x) as x approaches infinity, we know that the sine function oscillates between -1 and 1. Dividing sinx by any positive number x will approach 0. So, the limit of (sinx/x) as x approaches infinity is 0.
Now, add the limits of both terms:
lim (y) as x approaches infinity = lim (1 + (sinx/x)) as x approaches infinity
= 1 + 0
= 1
To find the limit as x approaches negative infinity, we can follow the same steps:
y = (x/x) + (sinx/x)
y = 1 + (sinx/x)
The limit of 1 as x approaches negative infinity is still 1.
To find the limit of (sinx/x) as x approaches negative infinity, we know that dividing sinx by any negative number x will still approach 0. So, the limit of (sinx/x) as x approaches negative infinity is 0.
Now, add the limits of both terms:
lim (y) as x approaches negative infinity = lim (1 + (sinx/x)) as x approaches negative infinity
= 1 + 0
= 1
Therefore, lim y as x approaches infinity is equal to 1, and lim y as x approaches negative infinity is also equal to 1.
2.) y = x / (1 + 2^x)
To find the limit as x approaches infinity, observe the denominator (1 + 2^x) grows exponentially with x.
As x increases, 2^x will tend to infinity, so the denominator (1 + 2^x) will also approach infinity.
Dividing any finite value x by a denominator that approaches infinity will yield a limit of 0. Therefore, as x approaches infinity, the limit of y will be 0.
lim y as x approaches infinity = 0
To find the limit as x approaches negative infinity, analyze the denominator (1 + 2^x) again.
When x is a large negative number, 2^x will tend to 0. Hence, the denominator (1 + 2^x) will approach 1.
Dividing any finite value x by a denominator that approaches 1 will result in a limit of x itself. Therefore, as x approaches negative infinity, the limit of y will be x.
lim y as x approaches negative infinity = x
In summary:
- For y = (x + sinx)/x, lim y as x approaches infinity and negative infinity is equal to 1.
- For y = x / (1 + 2^x), lim y as x approaches infinity is 0, and lim y as x approaches negative infinity is x.