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# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.26.6" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]Let $latex f$ be a nonnegative continuous function on $latex \Bbb R$ s.t $latex \int_0^{\infty}f(t)dt$ is finite then $latex \lim_{x \to \infty}f(x)=0$[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.26.6" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.26.6"]TIFR 2018 Part A, Problem 4 [/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.26.6" open="off"]Analysis [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.26.6" open="off"]Hard [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.26.6" open="off"]Real Analysis; Bartle and Sherbert [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.26.6" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.26.6"]Can you think of a function such that it has finite area but as $latex x$ goes to infinity, $latex f(x)$ goes to infinity?

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.26.6"]

Try to sketch the above function. Picture[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.26.6"]

See if the $latex lim f(x)$ exists then it must be zero right? but what if the limit fails to exist and the integral still gives you finite value? Hence, this is the case drawn in the above picture. Hence the answer is false generally.

# Watch the video

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# Similar Problems

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