Prove the statement using the 𝜀, 𝛿 definition of a limit. lim x → 1 7 + 2x3 = 3Given 𝜀 > 0, we need 𝛿 such that if 0 < |x − 1| < 𝛿, then 7 + 2x3 − 3  . But 7 + 2x3 − 3 < 𝜀 ⇔ 2x − 23 < 𝜀 ⇔ 23|x − 1| < 𝜀 ⇔ |x − 1| < . So if we choose 𝛿 = , then 0 < |x − 1| < 𝛿 ⇒ 7 + 2x3 − 3 < 𝜀. Thus, lim x → 1 7 + 2x3 = 3 by the definition of a limit.

if      0 < |x − 5| < 𝛿      then      5  <  𝜀that is,          if      0 < |x − 5| < 𝛿      then       <  𝜀5.This suggests that we should choose 𝛿 = 𝜀/5.2. Proof (showing that 𝛿 works). Given 𝜀 > 0, choose 𝛿 = 𝜀/5. If 0 <   < 𝛿, then|(5x − 7) − 18|  =   =  5  < 5𝛿 =  5   =  𝜀.Thusif    0 < |x − 5| < 𝛿    then    |(5x − 7) − 18| < 𝜀.Therefore, by the definition of a limitlim x → 5 5x − 7 = 18.

Consider the function f(x)={x2+34x+2839x+147ifx<−17ifx≥−17𝑓(𝑥)={𝑥2+34𝑥+283if𝑥<−179𝑥+147if𝑥≥−17.Step 1 of 3 :  Find limx→−17−f(x)lim𝑥→−17−⁡𝑓(𝑥).Answer

Suppose that h(x)={x2−𝑥+5 if 𝑥<25 if 𝑥=2𝑥3−1 if 𝑥>2Which of the following is equal to 7?I. lim⁡x→2−ℎ(𝑥)II. lim⁡x→2+ℎ(𝑥)III. lim⁡𝑥→2ℎ(𝑥)​ Suppose that h(x)= ⎩⎪⎨⎪⎧​ x 2 −x+55x 3 −1​  if  if  if ​ x<2x=2x>2​ Which of the following is equal to 7?I.  x→2−lim​ h(x)II.  x→2+lim​ h(x)III.  x→2lim​ h(x)​

Find the derivatives of the function 𝑓(𝑥) = −6𝑥2 − 9𝑥 − 7 by using the concept of limits.

Compute  lim⁡ 𝑥→3⁡1𝑥-13𝑥-3