SOLUTION: Evaluate using limit laws: Limit (as x approaches negative infinity) (x/sqrt(x^2-1)) Limit (as x approaches positive infinity) (x/sqrt(x^2-1))

Question 1200146: Evaluate using limit laws:
Limit (as x approaches negative infinity) (x/sqrt(x^2-1))
Limit (as x approaches positive infinity) (x/sqrt(x^2-1))
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

When x < 0,

When x > 0,

For very large positive or negative values of x, approaches .

This can be seen in the graph shown below
https://www.desmos.com/calculator/1llyy35ent
Desmos is a free graphing app.

The -1 at the end doesn't alter the radicand too much when x^2 is so very large.
Examples:
If x = 5, then x^2 = 25 and x^2-1 = 24
If x = 50, then x^2 = 2500 and x^2-1 = 2499
If x = 500, then x^2 = 250,000 and x^2-1 = 249,999
If x = 5000, then x^2 = 25,000,000 and x^2-1 = 24,999,999

This means approaches for large negative values of x.
Also, approaches for large positive values of x.

For very large negative values of x, approaches

For very large positive values of x, approaches

Therefore,

and

Verification using WolframAlpha
https://www.wolframalpha.com/input?i=x%2Fsqrt%28x%5E2-1%29+asymptotes

Verification using Desmos
https://www.desmos.com/calculator/hfq1eravms

RELATED QUESTIONS
Evaluate limit (as x approaches negative infinity)... (answered by ikleyn)

Limit of -(x^2+3)/(2-x)^2 as x approaches negative infinity (answered by solver91311)

Please help me with this problem: Evaluate the following limit: Limit of {{{(x +... (answered by Natolino1983)

Use the Squeeze Theorem to evaluate the limit as x approaches infinity (sinx)/x (answered by Fombitz)

as x approaches infinity find the limit of e^ (x-... (answered by stanbon)

Find the limit as x approaches infinity, if one exists... (answered by solver91311)

Provide an example of functions f(x) and g(x) with the limit as x approaches infinity of... (answered by richard1234)

What is the limit as x approaches infinity for arctan(3x^2-7x^4)... (answered by ikleyn)

As x approaches infinity, find limit of (x +... (answered by robertb)