Gate CS 2013 - 1 Mark Question
Duration: 3 min
This video lesson is available to enrolled students.
AI Summary
An AI-generated summary of this video lecture.
The video is an educational lecture on Calculus, specifically focusing on the continuity of functions. It begins with a title card displaying 'CALCULUS' over a background of mathematical formulas. The main content is a multiple-choice question from GATE CS 2013 asking to identify which function is continuous at x=3. The instructor, Yash Jain Sir, systematically evaluates four options (A, B, C, D) defined as piecewise functions. He explains the condition for continuity: the left-hand limit (LHL), right-hand limit (RHL), and the function value at the point must be equal.
Chapters
0:00 – 2:00 00:00-02:00
The video opens with a title card and then presents the problem statement. The instructor begins by analyzing Option A, a piecewise function defined as 2 if x=3, x-1 if x>3, and (x+3)/3 if x<3. To verify continuity, he writes annotations 'value', 'right', and 'left' next to the respective parts of the function. He calculates the function value at x=3 as 2. He then computes the right-hand limit (RHL) by substituting x=3 into (x-1), yielding 2. Similarly, he calculates the left-hand limit (LHL) by substituting x=3 into (x+3)/3, which equals 2. Seeing that LHL = RHL = Value, he circles the number 2 three times and places a checkmark next to Option A, confirming it is the correct answer. He then analyzes Option B, noting f(3)=4 but the limit is 5, leading him to cross it out. He moves to Option C, calculating LHL=6 and RHL=-1, and crosses it out due to inequality.
2:00 – 2:35 02:00-02:35
In the final segment, the instructor addresses Option D, which is defined as 1/(x^3 - 27) for x != 3. He points out that at x=3, the denominator becomes zero, implying the limit does not exist or is infinite. He marks this option as discontinuous, likely with an 'X' or by crossing it out, although the specific marking is brief. The instructor concludes the problem-solving session. The video ends with a black screen displaying the text 'THANKS FOR WATCHING' in a stylized white font. This final section reinforces the method of checking continuity by comparing limits and function values for all given options.
The lecture demonstrates a systematic approach to determining the continuity of piecewise functions at a specific point. By calculating the left-hand limit, right-hand limit, and the function value, the instructor verifies the condition LHL = RHL = f(c). The video effectively uses visual annotations to track these calculations for each option, eliminating incorrect choices until the correct function is identified.