Show that limx →1|x-1| sinx=0

Name: Problem 45, Chapter 2: Calculus: One Variable
Uploaded: 2021-08-07T08:37:25-08:00
Duration: 1 min 52 s
Channel: Umar Sohail Qureshi
Description: Show that limx →1|x-1| sinx=0

Key Concepts

Continuity

Continuity is a property that describes how functions behave without breaks or jumps at a specific point. For continuous functions like sine, the limit as the variable approaches any point is equal to the function's value at that point. This characteristic simplifies the evaluation of limits and supports the application of other limit laws.

Limit Laws and Rules for Products

Limit laws allow the computation of complex limits by breaking them down into simpler parts. Specifically, the product rule for limits states that if the limits of two functions exist, then the limit of their product is the product of those limits. This rule is especially helpful when one factor approaches zero, ensuring that the overall product also approaches zero.

Sine Function

The sine function is a periodic function that oscillates between -1 and 1. In calculus, the sine function is known for its continuity and smooth behavior, which facilitates the computation of limits. Its bounded nature means that when multiplied by a term that tends to zero, the product will typically tend to zero.

Absolute Value Function

The absolute value function gives the distance of a number from zero, which ensures that the output is always non-negative. This concept is important in understanding how distances and magnitudes are handled in limit computations, particularly when the sign of an expression could affect the outcome.

Limit of a Function

The limit of a function describes its behavior as the input approaches a particular value. This concept is central to calculus, as it underpins the definitions of continuity, derivatives, and integrals. Limits help in determining how functions behave near points of interest, even if the function is not defined at that point.

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