Question

7. Fill in the blanks: (a) Suppose $f(x)$ is positive for $x \ge a$. If $\lim_{t \to \infty} \int_a^t f(x) \, dx = L$, where $L$ is a __________, we say that $\int_a^\infty f(x) \, dx$ converges and $\int_a^\infty f(x) \, dx = L$. (b) $\lim_{x \to \infty} x^{1/4} = $ __________ (c) $\lim_{x \to \infty} x^{-1/4} = $ __________ (d) $\lim_{x \to 0^+} x^{-1/4} = $ __________ (e) $\lim_{t \to \infty} \int_1^t x^{-1/4} \, dx = $ __________

Name: 7 fill in the blanks a suppose fx is positive for x ge a if limt to infty intat fx dx l where l is a we say that intainfty fx dx converges and intainfty fx dx l b limx to inf 78022
Uploaded: 2021-07-25T06:08:05-08:00
Duration: 30 s
Channel: Adrian Co
Description: 7 fill in the blanks a suppose fx is positive for x ge a if limt to infty intat fx dx l where l is a we say that intainfty fx dx converges and intainfty fx dx l b limx to inf 78022

          7. Fill in the blanks:
(a) Suppose $f(x)$ is positive for $x \ge a$. If $\lim_{t \to \infty} \int_a^t f(x) \, dx = L$, where $L$ is a __________, we say that $\int_a^\infty f(x) \, dx$ converges and $\int_a^\infty f(x) \, dx = L$.
(b) $\lim_{x \to \infty} x^{1/4} = $ __________
(c) $\lim_{x \to \infty} x^{-1/4} = $ __________
(d) $\lim_{x \to 0^+} x^{-1/4} = $ __________
(e) $\lim_{t \to \infty} \int_1^t x^{-1/4} \, dx = $ __________

7 fill in the blanks a suppose fx is positive for x ge a if limt to infty intat fx dx l where l is a we say that intainfty fx dx converges and intainfty fx dx l b limx to inf 78022

Added by Michelle C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adrian Co

07/25/2021

Step 1

" By definition, if f(x) ≥ 0 on [a,∞) and lim_{t→∞} ∫_a^t f(x) dx = L exists as a finite real number, then ∫_a^∞ f(x) dx converges and equals L. Show more…

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7. Fill in the blanks: (a) Suppose $f(x)$ is positive for $x \ge a$. If $\lim_{t \to \infty} \int_a^t f(x) \, dx = L$, where $L$ is a __________, we say that $\int_a^\infty f(x) \, dx$ converges and $\int_a^\infty f(x) \, dx = L$. (b) $\lim_{x \to \infty} x^{1/4} = $ __________ (c) $\lim_{x \to \infty} x^{-1/4} = $ __________ (d) $\lim_{x \to 0^+} x^{-1/4} = $ __________ (e) $\lim_{t \to \infty} \int_1^t x^{-1/4} \, dx = $ __________