Problem 43 - Thomas Calculus

Question

Find the limits in Exercises \(37-48.\) $$\lim _{x \rightarrow 7} \frac{4}{(x-7)^{2}}$$

Step-by-Step Solution

Answer

The limit is positive infinity.

Step 1: Understand the function

The function given is \( f(x) = \frac{4}{(x-7)^2} \). It has a vertical asymptote at \(x = 7\) because the denominator becomes zero at this point.

Step 2: Examine the limit from the left

As \( x \) approaches 7 from the left \((x \to 7^{-})\), the term \((x-7)\) becomes a small negative number and \((x-7)^2\) becomes a small positive number. The fraction \(\frac{4}{(x-7)^2}\) thus approaches positive infinity.

Step 3: Examine the limit from the right

As \( x \) approaches 7 from the right \((x \to 7^{+})\), the term \((x-7)\) becomes a small positive number and \((x-7)^2\) stays as a positive number. Therefore, \(\frac{4}{(x-7)^2}\) approaches positive infinity.

Step 4: Conclude about the two-sided limit

Since both \(\lim_{x \to 7^{-}} \frac{4}{(x-7)^2}\) and \(\lim_{x \to 7^{+}} \frac{4}{(x-7)^2}\) are positive infinity, the two-sided limit \(\lim_{x \to 7} \frac{4}{(x-7)^2}\) is positive infinity.

Key Concepts

Vertical Asymptote

A vertical asymptote represents a place on a graph where a function approaches infinity as the input heads toward a certain value. This often occurs when the denominator of a fractional function zeroes out, causing the fraction itself to potentially become infinite.
In our example, the function is given by \( f(x) = \frac{4}{(x-7)^2} \). Here, the denominator \((x-7)^2\) equals zero when \(x = 7\). Hence, the function does not have a finite value at this point and displays a vertical asymptote.
Whenever you investigate a function and identify where the denominator equals zero, you've found a potential vertical asymptote. It's a handy way to visually comprehend how the function behaves near those critical points.

Two-Sided Limit

A two-sided limit expresses the behavior of a function as it approaches a particular input value from both sides. For instance, in the given function \( f(x) = \frac{4}{(x-7)^2} \), to find the two-sided limit at \(x = 7\), we need to observe the function from the left side (as x heads from values less than 7) and the right side (as x approaches from values greater than 7).
Here, both sides of the equation result in the same infinite behavior, which simplifies our analysis to conclude that the overall two-sided limit is positive infinity.
The understanding and calculation of two-sided limits are crucial since they help establish whether a function reaches a common value from both directions or diverges in different paths, aiding in identifying the bridge in continuity at that point.

Approaching from Left and Right

Approaching a point from the left and right entails seeing the function's progression towards a particular value separately from both sides.

• From the left: When \(x \to 7^{-}\), the variable \((x-7)\) forms a very tiny negative value. Squaring this yields a positive outcome, pushing \(\frac{4}{(x-7)^2}\) toward positive infinity.
• From the right: When \(x \to 7^{+}\), \((x-7)\) is a small positive number. Again, squaring retains positivity, leading \(\frac{4}{(x-7)^2}\) upwards to infinity.

In each case, the result is a movement towards positive infinity. This coordination from both approaches reflects the principle of limits, where the same result from both sides solidifies the value as \(x\) nears 7.