Problem 31
Question
$$ \lim _{x \rightarrow 1} \frac{x}{x-1}\\{\text { Ans. }+\infty,-\infty\\} $$
Step-by-Step Solution
Verified Answer
The limit of the function \(\frac{x}{x-1}\) as x approaches 1 is \(-\infty\) from the left side and \(+\infty\) from the right side.
1Step 1: Identify the function
The function we're working with is \(\frac{x}{x-1}\), and our task is to find the limit of this function as x approaches 1.
2Step 2: Analyze the denominator
In order to determine the limit value, we need to figure out what happens to the function as x approaches 1. Notice that as x gets closer to 1, the denominator (x-1) approaches 0. This means the fraction will approach either positive or negative infinity.
3Step 3: Determine the behavior of the function when x approaches 1
We can investigate the behavior of the function as x approaches 1 from both the left and the right sides:
- When x approaches 1 from the left side (x < 1), the denominator (x-1) is negative. Since the numerator (x) is also positive, the overall fraction will be negative. Therefore, the limit when x approaches 1 from the left side is \(-\infty\).
- When x approaches 1 from the right side (x > 1), the denominator (x-1) is positive. Since the numerator (x) is also positive, the overall fraction will be positive. Therefore, the limit when x approaches 1 from the right side is \(+\infty\).
4Step 4: State the final answer
Since the function approaches negative infinity when x approaches 1 from the left side, and positive infinity when x approaches 1 from the right side, the limit of the function as x approaches 1 is \(-\infty\) from the left side and \(+\infty\) from the right side.
Thus, the answer is:
Ans. \(+\infty\), \(-\infty\)
Key Concepts
One-sided limitsInfinite limitsBehavior of functions near discontinuities
One-sided limits
When we talk about one-sided limits in calculus, we're referring to the limit of a function as the input approaches a particular value from just one side—either from the left or the right. To understand one-sided limits, consider approaching a specific point on the x-axis and checking how the function behaves just before and just after reaching that point.
In the given exercise, the function is \[ \lim _{x \rightarrow 1} \frac{x}{x-1} \]. We examine the behavior as x approaches 1 from the left side (noted as \(x \to 1^-\), meaning less than 1) and from the right side (noted as \(x \to 1^+\), meaning greater than 1).
In the given exercise, the function is \[ \lim _{x \rightarrow 1} \frac{x}{x-1} \]. We examine the behavior as x approaches 1 from the left side (noted as \(x \to 1^-\), meaning less than 1) and from the right side (noted as \(x \to 1^+\), meaning greater than 1).
- For \(x \to 1^-\), when x is slightly less than 1, the denominator (x-1) is a small negative number. Consequently, the fraction becomes very negative and the one-sided limit is \(-\infty\).
- For \(x \to 1^+\), when x is slightly more than 1, the denominator (x-1) is a small positive number, making the fraction very positive. Thus, the one-sided limit is \(+\infty\).
Infinite limits
Infinite limits occur when the value of a function becomes infinitely large as the input approaches a specific point. These can either be positively infinite, meaning the function grows without bounds, or negatively infinite, where the function decreases without bounds.
In our exercise, as x gets closer to 1, the denominator (x-1) tends towards zero. However, since the numerator x remains close to a fixed value (1), the result is that our function \( \frac{x}{x-1} \) tends towards either positive or negative infinity depending on the direction of approach.
In our exercise, as x gets closer to 1, the denominator (x-1) tends towards zero. However, since the numerator x remains close to a fixed value (1), the result is that our function \( \frac{x}{x-1} \) tends towards either positive or negative infinity depending on the direction of approach.
- If x approaches 1 from the left (\(x \to 1^-\)), the function trends towards \(-\infty\)
- If x approaches 1 from the right (\(x \to 1^+\)), the function trends towards \(+\infty\)
Behavior of functions near discontinuities
Discontinuities in a function tell us that the function makes a leap or has a break in its graph at a certain point. Understanding the behavior of functions near these discontinuities helps in predicting how a function might behave overall. At a discontinuity, limits might not exist, but one-sided and infinite limits can still give insights.
For the function \( \frac{x}{x-1} \), there's a discontinuity at \(x=1\). As x approaches 1, the denominator approaches zero, causing the function value to spike aggressively either towards positive or negative infinity, depending on the direction.
For the function \( \frac{x}{x-1} \), there's a discontinuity at \(x=1\). As x approaches 1, the denominator approaches zero, causing the function value to spike aggressively either towards positive or negative infinity, depending on the direction.
- As \(x \to 1^-\), the function races to \(-\infty\)
- As \(x \to 1^+\), it soars to \(+\infty\)
Other exercises in this chapter
Problem 29
$$ \lim _{x \rightarrow 0}(1+x)^{\sin x} \quad\\{\text { Ans. } 1\\} $$
View solution Problem 30
$$ \lim _{x \rightarrow 0}\left(\cos ^{-1} x\right)^{\cos x}\left\\{\text { Ans. } \frac{\pi}{2}\right\\} $$
View solution Problem 32
$$ \lim _{x \rightarrow 0} \ln x+e^{x}\\{\text { Ans. }-\infty\\} $$
View solution Problem 33
$$ \lim _{x \rightarrow \frac{\pi}{2}} \tan x+\tan 2 x \quad\\{\text { Ans. }-\infty,+\infty\\} $$
View solution