Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.
lim x→0 x^2=0 (Hint: Use the identity √x2=|x|.)

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Start by recalling the precise definition of a limit: For every $ \varepsilon > 0 $, there exists a $ \delta > 0 $ such that if $ 0 < |x - 0| < \delta $, then $ |x^2 - 0| < \varepsilon $.

Simplify the expression $ |x^2 - 0| $ to $ |x^2| $, which is equal to $ |x|^2 $.

To ensure $ |x|^2 < \varepsilon $, we need $ |x| < \sqrt{\varepsilon} $.

Choose $ \delta = \sqrt{\varepsilon} $. This choice of $ \delta $ ensures that whenever $ 0 < |x| < \delta $, it follows that $ |x|^2 < \varepsilon $.

Conclude that the relationship $ \delta = \sqrt{\varepsilon} $ guarantees the limit $ \lim_{x \to 0} x^2 = 0 $ exists.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit Definition

The precise definition of a limit states that for a function f(x) to approach a limit L as x approaches a value a, for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε. This formalism is crucial for proving limits rigorously.

Epsilon-Delta Relationship

In the context of limits, the ε (epsilon) represents how close f(x) must be to the limit L, while δ (delta) represents how close x must be to the point a. Establishing a relationship between ε and δ is essential to demonstrate that as x gets sufficiently close to a, f(x) will be within ε of L.

Absolute Value and Squaring

The identity √x² = |x| is important in this limit proof because it allows us to express x² in terms of its absolute value. This is particularly useful when dealing with limits approaching zero, as it simplifies the analysis of the function's behavior near that point.

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Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).