Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each limit. $$\begin{array}{l} \text { (a) } \lim _{x \rightarrow 0^{-}} \log |x| \\ \text { (b) } \lim _{x \rightarrow 0^{+}} \log |x| \\ \text { (c) } \lim _{x \rightarrow 0} \log |x| \end{array}$$

Step-by-Step Solution

Answer

All limits evaluate to \(-\infty\).

Step 1: Interpret the Limit (a)

For the limit \( \lim_{x \rightarrow 0^{-}} \log |x| \), we are approaching zero from the left side. Here, \( |x| = -x \) since \( x \) is negative and close to zero. The expression becomes \( \log(-x) \) as \( x \rightarrow 0^{-}. \)

Step 2: Evaluate Logarithm (a)

As \( x \rightarrow 0^{-} \), \( -x \) approaches zero from the positive side. The logarithm \( \log(-x) \) heads towards \( -\infty \) because the logarithm of a number approaching zero from the positive side is negative infinity.

Step 3: Conclusion for Limit (a)

Thus, \( \lim_{x \rightarrow 0^{-}} \log |x| = -\infty. \)

Step 4: Interpret the Limit (b)

Now consider \( \lim_{x \rightarrow 0^{+}} \log |x| \). Approaching zero from the right, \( |x| = x \) as \( x \) is positive. So, the expression is \( \log(x) \) as \( x \rightarrow 0^{+}. \)

Step 5: Evaluate Logarithm (b)

As \( x \rightarrow 0^{+}, \log(x) \) also tends towards \( -\infty \) because the logarithm of a number approaching zero from the positive side is negative infinity.

Step 6: Conclusion for Limit (b)

Thus, \( \lim_{x \rightarrow 0^{+}} \log |x| = -\infty. \)

Step 7: Interpret the Two-sided Limit (c)

For \( \lim_{x \rightarrow 0} \log |x| \), both left-hand and right-hand limits need to be considered. Since both are approaching \( -\infty \), the two-sided limit will be the same as the one-sided limits.

Step 8: Conclusion for Limit (c)

Therefore, \( \lim_{x \rightarrow 0} \log |x| = -\infty. \)

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. When you see \( \log|x| \), it refers to the logarithm with base 10 (common logarithm) of the absolute value of \( x \).

• If \( x \) is positive, \( \log|x| = \log x \).
• If \( x \) is negative, \( \log|x| = \log (-x) \) because the absolute value negates the negative sign.

The most important property here is how logarithms behave as their arguments approach zero. Whether you approach from the positive or negative side, the closer \( |x| \) gets to zero, the more the logarithmic function approaches negative infinity, because you cannot take the logarithm of zero or a negative number without employing complex numbers.

One-Sided Limits

One-sided limits refer to the behavior of a function as the input approaches a specified point from either the left (negative side) or the right (positive side). For \( \lim_{x \rightarrow 0^{-}} \log |x| \), we understand this as approaching zero from the negative side. For \( \lim_{x \rightarrow 0^{+}} \log |x| \), this means approaching zero from the positive side. These are essential in cases where the function behaves differently on either side of the point.
For example:

• From the left (negative side), the expression \( \log(-x) \) approaches \( -\infty \) since \( -x \) becomes a small positive value approaching zero.
• From the right (positive side), the expression \( \log(x) \) also approaches \( -\infty \) as \( x \) heads towards zero from positive values.

Both limits being minus infinity indicates that as \( x \) tends to zero from either direction, the function exhibits a vertical asymptote.

Two-Sided Limits

Two-sided limits involve examining what a function approaches as the input variable gets closer to an approaching value from both sides. If both the one-sided limits exist and are equal, the two-sided limit will exist and equal these limits. In the case of \( \lim_{x \rightarrow 0} \log |x| \), we consider both:

• \( \lim_{x \rightarrow 0^{-}} \log |x| = -\infty \)
• \( \lim_{x \rightarrow 0^{+}} \log |x| = -\infty \)

Here, since both one-sided limits are \( -\infty \), the two-sided limit is also \( -\infty \). This reflects that as you approach zero, regardless of direction, the logarithmic function is not bounded and heads towards negative infinity. Understanding these limits helps visualize how the function behaves and recognize discontinuities or asymptotes.