# Thread: Help with Trig Limits

1. ## Help with Trig Limits

I understand the basicsof trig limits, but of a gamut of 500 questions there are several I just can't get the answer for. What am I doing wrong?

1. Lim (sinx / x+tanx)
x->0

2. Lim([cot^2]x / 1-cscx)
x->pi/2

3. Lim (tan4x/tan7x)
x->0

1. I tend to end up at a mess here: I multiply the top and bottom by 'x' so I can use the sinx/x identity, and ultimately end up with (xcosx/(xcosx+sinx)) and get lost from there. According to the back of the book, the answer is 1/2.

2. Everything I try here just ends up a mess

3. I tend to end up at

(sin4x)/4x . 7x/(sin7x) . 4x/(cos4x) . (cos7x/7x)
=(4/7) . (0) . (undefined)
the answer in the back of the book is 4/7,and I don't know how to get there

Thanks in advance! I have a test Tuesday,and I'm so lost in all of this.

2. Originally Posted by Futant
I understand the basicsof trig limits, but of a gamut of 500 questions there are several I just can't get the answer for. What am I doing wrong?

1. Lim (sinx / x+tanx)
x->0

2. Lim([cot^2]x / 1-cscx)
x->pi/2

3. Lim (tan4x/tan7x)
x->0

1. I tend to end up at a mess here: I multiply the top and bottom by 'x' so I can use the sinx/x identity, and ultimately end up with (xcosx/(xcosx+sinx)) and get lost from there. According to the back of the book, the answer is 1/2.

2. Everything I try here just ends up a mess

3. I tend to end up at

(sin4x)/4x . 7x/(sin7x) . 4x/(cos4x) . (cos7x/7x)
=(4/7) . (0) . (undefined)
the answer in the back of the book is 4/7,and I don't know how to get there

Thanks in advance! I have a test Tuesday,and I'm so lost in all of this.
for 1 )
$\displaystyle \frac{\sin(x)}{x+\tan(x)}=\frac{\sin(x)}{x+\frac{\ sin(x)}{\cos(x)}}=\frac{\sin(x)}{\sin(x)}\cdot \frac{1}{\frac{x}{\sin(x)}+\frac{1}{\cos(x)}}$ Now as $\displaystyle x \to 0$ we get $\displaystyle \frac{1}{1+1}=\frac{1}{2}$

for 2) use the identity that $\displaystyle 1+\cot^2(x)=csc^2(x)$ and factor and reduce.

for 3) multiply the top and bottom by $\displaystyle \frac{4\cdot 7 \cdot x}{4 \cdot 7 \cdot x}$ and do what you tried. (you have an extra 4x)