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1. Find the following limits, if they exist, justify your answers fully. ([x] denotes the integer part of x).
a. [tex]\[lim_{x \to 1+} \frac{[x]}{x^2}[/tex]
b.[tex]\[lim_{x \to 1-} \frac{[x]-x}{[x^2]-x^2}[/tex]
c.[tex]\[lim_{x \to \infty}[/tex](1/x)^(1/x)
d.[tex]\[lim_{x \to 0} \frac{exp(2x)-1}{ln(1+x)}[/tex]
e.[tex]\[lim_{x \to 0} \frac{x^2sin(1/x)}{sin(x)}[/tex]
3. The Attempt at a Solution .
I am really stuck on the first two and don't know how to start, I am confused about how to deal with the integer parts in the limits. Could someone please offer some guidance?
For d, I used L Hopitals rule and got a limit of 2, is this correct? For e, I tried to use L Hopitals rule and think I managed to show that the limit was the same as the limit of 2sin(1/x) -cos(1/x)/x^3 and hence that it does not exist. Have I done these right?
a. [tex]\[lim_{x \to 1+} \frac{[x]}{x^2}[/tex]
b.[tex]\[lim_{x \to 1-} \frac{[x]-x}{[x^2]-x^2}[/tex]
c.[tex]\[lim_{x \to \infty}[/tex](1/x)^(1/x)
d.[tex]\[lim_{x \to 0} \frac{exp(2x)-1}{ln(1+x)}[/tex]
e.[tex]\[lim_{x \to 0} \frac{x^2sin(1/x)}{sin(x)}[/tex]
3. The Attempt at a Solution .
I am really stuck on the first two and don't know how to start, I am confused about how to deal with the integer parts in the limits. Could someone please offer some guidance?
For d, I used L Hopitals rule and got a limit of 2, is this correct? For e, I tried to use L Hopitals rule and think I managed to show that the limit was the same as the limit of 2sin(1/x) -cos(1/x)/x^3 and hence that it does not exist. Have I done these right?