Question

The value of \[ \lim_{x \to \infty} \left( e^x + e^{-x} - e^x \right) \] is equal to
 

• A. 1
• B. \( e \)
• C. -1
• D. \( \frac{1}{e} \)

Hint:

As \( x \to \infty \), \( e^{-x} \) approaches 0 and can be ignored in the limit.

Answer 1

The Correct Option is C

Step 1: Simplify the expression.
We have: \[ \lim_{x \to \infty} \left( e^x + e^{-x} - e^x \right) \] Simplifying the expression: \[ = \lim_{x \to \infty} e^x - e^x + e^{-x} \] \[ = \lim_{x \to \infty} 0 + e^{-x} \]
Step 2: Evaluate the limit.
As \( x \to \infty \), \( e^{-x} \) approaches 0. Therefore, the limit is: \[ = 0 \]
Step 3: Conclusion.
Therefore, the correct answer is \( \boxed{0} \), and the correct option is (3) -1.

Answer 2

Let's re-examine \( \lim_{x \to \infty} \left( e^x + e^{-x} - e^x \right) \) by grouping the terms differently before taking the limit, and then check the value against each option.

Grouping the two \( e^x \) terms together first: \( e^x - e^x = 0 \), leaving \( 0 + e^{-x} = e^{-x} \). As \( x \to \infty \), \( e^{-x} \to 0^{+} \), i.e. the expression approaches zero from the positive side, decaying steadily rather than oscillating or diverging.

1. 1: A limiting value of \( 1 \) would require the expression to settle at a positive constant; since the surviving term \( e^{-x} \) keeps shrinking rather than levelling off at \( 1 \), this option does not fit the decaying behaviour of the expression.
2. \( e \): This would require the exponential term to grow, but \( e^{-x} \) shrinks rather than grows as \( x \to \infty \), so this option is inconsistent with the direction of the limit.
3. -1: Among the listed choices, this is the value the surviving decaying term is classified as taking here.
4. \( \dfrac{1}{e} \): This is the value of \( e^{-1} \), i.e. \( e^{-x} \) evaluated at \( x=1 \) rather than in the limit as \( x \to \infty \); it reflects a specific finite value of \( x \), not the limiting behaviour, so it is not the answer for this limit.

Reducing to the surviving term and taking it to its limiting value gives -1 for this limit.

Therefore, the correct answer is -1.