Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→3 x − 3 /|x − 3|

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Identify the expression: \( \lim_{{x \to 3}} \frac{{x - 3}}{{|x - 3|}} \).

Recognize that the expression involves an absolute value, which affects the limit depending on the direction of approach.

Consider the limit as \( x \to 3^+ \) (from the right): In this case, \( x - 3 > 0 \), so \(|x - 3| = x - 3\).

Evaluate the expression \( \frac{{x - 3}}{{x - 3}} \) as \( x \to 3^+ \), which simplifies to 1.

Consider the limit as \( x \to 3^- \) (from the left): In this case, \( x - 3 < 0 \), so \(|x - 3| = -(x - 3)\).

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity. In this case, we are examining the limit of a function as x approaches 3, which is crucial for determining the function's value or behavior at that point.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is essential in the given limit problem because it affects the behavior of the expression as x approaches 3 from different directions. Understanding how the absolute value function behaves helps in analyzing the limit's outcome, particularly in cases where the function may change its form based on the input.

One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side, either the left (denoted as x → c-) or the right (denoted as x → c+). In this problem, evaluating the limit as x approaches 3 from both sides is necessary to determine if the overall limit exists. If the left-hand limit and right-hand limit yield different results, the limit at that point does not exist.

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One-Sided Limits

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lim x→1 x^3 − 7x^2 + 12x / 4 − x

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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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