What is the limit as x approaches negative infinity for x + sqrt(x^2+3)?

In summary, the given limit is equivalent to the limit of the expression 3 over the square root of x squared plus 3 plus x as x approaches infinity, which is a trivial limit.
  • #1
dnt
238
0

Homework Statement



limit as x-> -oo of x+(x^2+3)^(1/2) (square root)

Homework Equations



n/a

The Attempt at a Solution



i first multiplied the top and bottom (which is just 1) by the conjugate to get:

-3
--------------
[x - (x^2+3)^(1/2)]

then i divided by x on top and bottom to get:

-3/x
-----------
[x/x - (1 + 3/x2)^(1/2)]

but now what do i do? the top goes to 0 but the bottom also goes to 0 (because its 1 - (1+0)^(1/2)). but the answer is suppose to be 0.

0/0 isn't 0. can someone point me in the right direction? thanks.
 
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  • #2
[tex] \lim_{x\rightarrow -\infty} x+\sqrt{x^{2}+3} =\lim_{x\rightarrow \infty}
\sqrt{x^{2}+3}-x [/tex]

After multiplying by the conjugate expression you're facing the limit

[tex] \lim_{x\rightarrow \infty} \frac{3}{\sqrt{x^{2}+3}+x} [/tex]

which is trivial.

Daniel.
 

FAQ: What is the limit as x approaches negative infinity for x + sqrt(x^2+3)?

What does "limit to negative infinity" mean?

The phrase "limit to negative infinity" refers to the behavior of a mathematical function as its input approaches negative infinity. In other words, it describes what happens to the output of the function as the input becomes more and more negative.

How is the limit to negative infinity different from the limit to positive infinity?

The limit to negative infinity is the opposite of the limit to positive infinity. While the limit to negative infinity describes the behavior of a function as the input approaches negative infinity, the limit to positive infinity describes the behavior as the input approaches positive infinity.

Can a function have a limit to negative infinity?

Yes, a function can have a limit to negative infinity. This means that as the input of the function approaches negative infinity, the output of the function will also approach a specific value. However, it is possible for a function to have a limit to negative infinity and not to positive infinity, or vice versa.

How is the limit to negative infinity calculated?

The limit to negative infinity is calculated using the same principles as other limits. The input of the function is substituted with increasingly negative values, and the corresponding output values are observed. If the output values approach a specific value as the input becomes more and more negative, then that value is the limit to negative infinity.

What is the significance of the limit to negative infinity in calculus?

The limit to negative infinity is an important concept in calculus as it helps us understand the behavior of a function at extreme values. It is also a key component in determining the continuity and differentiability of a function, as well as in evaluating integrals and solving differential equations.

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