Find the limit. Use a graphing utility to verify your result. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.)
[tex]\lim _{x \rightarrow-\infty}\left(4 x+\sqrt{16 x^2-x}\right)[/tex]
Answer (2)
Rationalize the numerator of the expression by multiplying by the conjugate.
Simplify the expression by eliminating the square root in the numerator.
Divide both numerator and denominator by x , considering that x approaches negative infinity.
Evaluate the limit by substituting x = − ∞ , which yields 8 1 .
Explanation
Problem Setup We are asked to find the limit of the expression 4 x + 16 x 2 − x as x approaches − ∞ . We will use the hint to rationalize the numerator.
Rationalizing the Numerator Multiply the expression by 4 x − 16 x 2 − x 4 x − 16 x 2 − x :
x → − ∞ lim ( 4 x + 16 x 2 − x ) = x → − ∞ lim ( 4 x + 16 x 2 − x ) ⋅ 4 x − 16 x 2 − x 4 x − 16 x 2 − x
Simplifying the Expression Simplify the numerator:
x → − ∞ lim 4 x − 16 x 2 − x ( 4 x ) 2 − ( 16 x 2 − x ) 2 = x → − ∞ lim 4 x − 16 x 2 − x 16 x 2 − ( 16 x 2 − x ) = x → − ∞ lim 4 x − 16 x 2 − x x
Rewriting with Absolute Value Since x → − ∞ , x < 0 . Therefore, x 2 = ∣ x ∣ = − x . We can rewrite 16 x 2 − x as 16 x 2 ( 1 − 16 x 1 ) = 4∣ x ∣ 1 − 16 x 1 = − 4 x 1 − 16 x 1 .
x → − ∞ lim 4 x − 16 x 2 − x x = x → − ∞ lim 4 x − ( − 4 x ) 1 − 16 x 1 x = x → − ∞ lim 4 x + 4 x 1 − 16 x 1 x
Dividing by x Divide both the numerator and the denominator by x :
x → − ∞ lim 4 x + 4 x 1 − 16 x 1 x = x → − ∞ lim 4 + 4 1 − 16 x 1 1
Evaluating the Limit Evaluate the limit as x → − ∞ :
x → − ∞ lim 4 + 4 1 − 16 x 1 1 = 4 + 4 1 − 0 1 = 4 + 4 1 = 8 1
Final Answer Therefore, the limit is 8 1 .
Examples
In physics, when analyzing the motion of objects under certain forces, you might encounter expressions similar to the one in this problem. For example, when dealing with asymptotic behavior in potential energy calculations or when studying the relativistic effects at very high speeds, rationalizing and finding limits helps simplify complex equations and understand the system's behavior as variables approach extreme values. This skill is crucial for making accurate predictions and designing experiments.
The limit of the expression as x approaches negative infinity is 8 1 . This is derived by rationalizing the numerator and simplifying the expression. Finally, substituting in the limit leads to the answer.
;
